Tell your mom you're going to get ready at a friends house, and that you and a few people are just going as friends. Buy a dress if you have to (sth super cheap) and then try to find some money for a tux (I don't know if you have your own source of money? or if you could scrounge some up some way before prom). If this girl is understanding of your situation she may be on board with understanding that you come to her house in a dress and change. Your mom may also understand that you might want to change into your dress later on, and let you wear your normal clothes and just bring the dress along with you. Throw a suit in the same bag (doesn't really have to be a tux does it, though I guess it'd be nice), and just change into that instead. If your mom asks for pictures or anything... well who cares its already over and you had your great night.
I didn't go to prom. Didn't want to though. I went to a sick music festival instead =] me and my friend had the time of our lives, while our other friends had a not-so-great time at prom. Literally we felt bad talking about the show because I'm pretty sure they got busted drinking or smoking or both. Its not a decision I regret, but it sounds like one you might.