Is there a way to get a free house in skyrim?
You can get a free house in Solstheim if you have Dragonborn installed and you stop an assassination.
You could squat in the Abandoned_House or one of the other safe places.
SPOILERS AS FOLLOWS
The only one I can think of is the ArchMage Hall in the College of Winterhold. It is free (you must complete a quest line to get it though), but it is not really a house, more of a huge room, many chests, an ingredient garden (alchemy) a bed, and lots of loot and potions. It also has an alchemy table and an enchanting table (plus lots of soul gems). It has the same setting of a house, and is even bigger/better than most.
Also, it does not require any DLC to obtain it.
There is a glitch that allow you to get a house in town for free when becoming Thane:
Talk to the steward about the house, say yes and then quickly go to the nearest container and place all of your gold before the steward finishes the dialogue.
You should get the book and key, now take the gold back from the container. NOTE: In all the city palaces, make sure to find the nearest container and use Slow Time shout if you need help.
Here are some near containers in Cities
Markarth: Dwarven chest next to the steward.
Solitude: Cupboard next to Falk.
Whiterun: Drawers in the courtroom [Use Slow Time or Whirlwind Sprint], chest in the maproom, and other containers in steward's bedroom.
Windhelm: Containers in kitchen [Use Slow Time or Whirlwind Sprint].
Riften: Chests left and right of the court.