i bought a couple of those U-shaped hooks for hanging bikes, but my room is like 6x8 ft^2. i don't have a lot of space to play with, so would you think that it would damage my bike at all to just hang it by the front wheel?

I dont think any damage should be done to it. If you're worried they make hooks that are just for hanging it by the front wheel. They're about $15, Supergo always has sales on them for $10.

In most cases, it doesn't matter. Once you climb onto your bike and start to ride, that front wheel has to handle forces way greater than the weight of your bike just hanging statically.

If you have a mountain bike that uses an oil-filled suspension, a FEW of those don't take kindly to hanging at odd angles - they'll leak.

If you are picky about esthetics, keep an eye on the plastic coating on the hook. It can wear away and allow the metal hook to leave cosmetic marks on your rim.

I hang all mine that way (I add a 2" piece of rubber gas line as added cushion)

But from the back wheel and with the front wheel removed. The forks rest against a chair rail I added to the wall.

i made my own hook...works great