There was an Arena in Morrowind, but not the same as the one in Oblivion. Oblivion was Roman style gladiator games. Morrowind was a place where people could legally fight to the death in answer to a challenge.
That said, it wouldn't surprise me to see some form of an arena in Skyrim. I don't care one way or the other. Aside from achievments, a couple random quests and money the arena in Oblivion didn't have much to offer.
It was convenient for that. The easiest money though IMO was making potions. I stole every food item I found and made tons of potions. I'd keep the useful ones including poisons and restoration potions, and would sell the junk ones. Sometimes i'd buy all
the ingredients in a shop, make as many potions as I could and then sell them back to the shop owner for more than I paid to start. This was also beneficial for increasing my alchemy skills.
Correct me if I'm wrong, but can't you and other NPCs be challenged to duels? If you or someone else makes the challenge then you can legally fight them to the death in public. I remember reading about that months back.