I do this all the time, then edit the videos and put them on YouTube to have friends review. Sometimes they catch things my instructor didn't, or they have some pointers that I've never been told. It's extremely helpful to video tape yourself for review, or just to see how you're progressing over time.
What I use is a cheap waterproof video camera with a flexible tripod. The tripod actually wraps around the fencing for the arena and I have it mounted in the corner pointing towards the center for maximum coverage. Don't use a fancy camera, the dust will kill it pretty quickly... you can buy a cheap Kodak or FlipHD pocket camera for under $80 plus tripod. It's pretty good quality and very rugged, so I don't worry about it ever getting damaged.