Pictures would help but yes you can. One can cut the whole rosette of leaving stem below it and let it callous for a while and then put it in a fast draining mix and mist it till it has roots. water then can commence. It needs roots to absorb water, so watering before it has roots just encourages rot. The aloe has enough energy stored in its leaves to keep it alive till it makes roots. This can take awhile. Now is a good time since there is warmth most places north of the Equator.

I had a big gold tooth aloe that I cut in the fall, then just sat it at the edge if another pot in the window... I didn't actually put it in medium until almost spring. Thats a little extreme, but as wanton said, let it dry really well first.
It will look a little shriveled before it roots well, and that's ok. Plus, if you leave the base alone, it may sprout out little aloe shoots as well.
Good luck
Chris

Wishbone sent me a photo through my email and it is either an Aloe X nobiles (goldtoothed aloe) or an Aloe distans (also called a gold toothed aloe). My nobiles is a huge pupper so it might be an Aloe distans that grows a stalk and then lies on the ground (in nature) and crawls around the place. Her picture showed no pups so maybe it is an Aloe distans. If left to its devises it should succker along its stem also, and create a radiating clump of stemmed aloes.. Maybe it needs to be potted up so the roots have more room. I suggest googling both these aloes names and doing some research before you start whacking. If you do whack . Keep the base it might send out multiple heads in time. What you are seeing is probably a aloe in its clumsy period of life.... like adolescence or something.

Thanks! I haven't done anything yet....

Yes, you can. Attached is the Aloe FAQ link, and after you cut, follow the directions there for rerooting. A picture of your plant would help, too.

Here is a link that might be useful: Aloe FAQ