Is writing a letter to ask someone out a bad idea?

I attended a performance at a small local theatre over the holiday weekend. One of the actresses just grabbed my attention. I was wondering if it would be socially acceptable to write a letter and mail it to the theatre in order to ask the actress out to lunch. I don't have another way of talking to her and I was afraid that hanging out till she left the back stage area would be a little too much. I could use any help anyone can offer as to what I should do or not do.

It would surprise me if she accepted such an invitation from someone she doesn't know. If you decide to try that, inviting her to bring a friend would improve your chances and including a dozen roses with the invitation might help too.