I dont think a colluseum like arena would fit skyrim...
it certainly fits the capital city of the land.
but skyrim should have something more underground and gritty.. like that first arena in the desert you see in the movie gladiator.
Or a barroom (mead-hall?) fight circuit, where you go around having semi-legal matches in taverns. It's more underground, more wild, and more frontiersy.
But I do think that an arena type competition would not necessarily be out of place in Skyrim.
What might be even better than that, though, would be a more broad and bigger type of contest that included hunting, fighting, survival and endurance. You would have a guild to organize "matches" and the object would be to go out into the wild and kill a particular beast or enemy first, or an actual fight, or a race. It seems like this would better suit the rugged, outdoorsy spirit of the Nords.