I'm generally fairly easy-going when it comes to fictional characters. After all, we don't know the sexuality of anyone in the game for certain. People in happy, loving, opposite-sex relationships might identify as bisexual rather than heterosexual, just like in real life. Also, there are an awful lot of characters who aren't currently in relationships, and the game doesn't have speech options for you to quiz them about their sexual orientation. The best you can do is put on an Amulet of Mara and ask people if they're interested in marrying you. The same 60 or so characters are available for marriage, regardless of the gender or race of the character you're playing. And if they have a high enough relationship level with you, they will always say yes.
So Skyrim's a lot more generous than many games in terms of what it allows the player. Opposite-sex and same-sex relationships are treated absolutely identically. But the only characters who talk about being romantically involved are people in opposite-sex relationships. There may be same-sex relationships, but they're not overt.
The only overt same-sex couple are both dead.
The tale of Hrodulf and Bjornolfr is a tragedy which you only piece together if you have enough interest to do so. Wandering the island of Solstheim, which was added by the Dragonborn expansion pack, you come across a burned-out shack called "Hrodulf's house". When you go into the basement, you hear bandits reading out a love letter from Bjornolfr to Hrodulf and laughing at it.
Bandit 1: Listen to this one! "My nights have been impossible to bear without you." "I beg of you: Please depart that blasted isle and return to Solitude at once!"
Bandit 2: Ha ha ha! No more! What a fool.
Bandit 1: Do you think he's the one we left in the tunnel?
Bandit 2: Who cares? Nords all look the same to me.
Bandit 1: Ha! Wait... Did you hear that? Sounds like those Ash Spawn are back.
Bandit 2: Well, at least if the other Nord returns, they'll take care of him.
You can kill the two bandits, and then take the letter from the table. Exploring the area you find Hrodolf's journal, which shows that he was being driven insane by strange noises from under his house. When you move the bookshelf (the clue to do so is both in the journal, and also from the books which are all over the floor), you enter a rough tunnel which has been dug recently - there is a pickaxe, torch, and shovel inside. At the end of the tunnel is some Dwemer machinery - lots of pipes made of some copper-coloured metal moving up and down with clanking noises, and hissing steam. (Think steampunk). There is also a dead man, who turns out to be Bjornolfr. His weapon is dropped on the ground, and he's holding a letter from Hrodulf.
Hrodolf himself turns out to be dead on the ground a little way south of his house, and this is where the real tragedy begins. He was killed presumably by the Burnt Spriggan (a type of fire monster) who's standing over him, and you have to fight that monster in order to look at him properly. When you do so, you find that he's lying dead by his boat with a broken war axe (when weapons can't even break in the game) and all his worldly possessions scattered around. There's a chest, a satchel, and a strongbox. Then to really add pathos, inside his boat is an Amulet of Mara and a gold diamond ring. It's as if having reached the end of the tunnel and found what was making the noise, he came to his senses and was going to sail back to Bjornolfr on the Skyrim mainland - and ask him to marry him. Remember I said the Amulet of Mara is what you wear if you want to ask someone about marriage? In the meantime, Bjornolfr was so alarmed by Hrodolf's weird letters that he left Solitude and came to Solstheim to get him, and somehow missed him entirely, getting killed by bandits instead.
It's just so sad. And you can't even reunite the lovers in death because Hrodolf was outside while Bjornolfr was in the basement, and while you can pick up corpses and carry them, you can't carry them through doors. Wanting to commemorate them somehow, Martin, my character, took their weapons and letters and the Amulet of Mara and gold diamond ring back to his house, and arranged them together in a display case. Don't ask how long that took. Skyrim display cases are evil. He would have preferred to take them to a Shrine of Mara and arrange them there, but they would have disappeared after 10 days or so. This was a compromise based on game mechanics.
Anyway. I don't know what to think. On the one hand, this is far more overt than the only same-sex couple in the whole of Oblivion (the previous game in the Elder Scrolls series). Their relationship is alluded to so vaguely that several people have argued they were only good friends. Again, you only find out about their relationship after they are both dead, but at least you get to meet them as people before they die. On the other hand, Oblivion was released in 2006. Skyrim was released five years later, in 2011 - and it does seem wrong to me that a game that is rated at the very least "teen" in all regions, and "mature" in most, doesn't have a few living same-sex couples.