I use a 17-40 f/4L, but that's way out of your price range. If you want zoom, you'll most likely ahve to go with a kit lense. Otherwise, you could get a fairly nice prime for that price. I don't know too much about primes yet, so I'll let somebody else tell you about those.
Wide angle lenses can be very useful for dynamic angles in skiing photos, but you can film skiing with not so wide lenses because you have the option of camera movements throughout your shots.
Getting something in the 20-35 range would be good for most everything you want to do. I don't much but don't think anything wider than about 20mm would be useful for me for filming skiing on a crop sensor camera.
What other lenses have you used before? I wouldn't make a decision based off of one lens you've used a few times. Buying a lens should be a very big decision since you're lens will most likely last longer than you body.