Should I DM her?

I'm a 18 year old guy and I'm in my last month of my senior year. My senior prom is coming up, and me and my best friend (girl) decided that we would go together. However, now (2 days after tickets are no longer being sold) I began to talk to this girl in my psychology class. She is really cute and I really enjoy talking to her while in the class. I can't ask for her number in class however because she is never alone and she is always with her friend.

My question is should I DM the girl just to start talking to her? I don't want it to be awkward because I'm going to prom with another girl but I'm trying to go out with her. And if you think I should DM her how should I do it without it being too awkward,

Thank you for any answer that you could give.