Can i ask her out on facebook although we never chatted before ?

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So some months ago I met by chance a girl which used to be in my class a few years ago. We had a little chit-chat and then when I got home I searched for her on facebook and added her to my friends.
Lately I've been thinking that it'd be nice if we hung out but the problem is that we're never online at the same time, so we never had any conversation via facebook. I don't want to wait for the moment when she and i will be online so we can chat a bit and then I propose to her to meet somewhere, it'd take too long. Take note that facebook is the only means of communication I have, I don't have her number or e-mail or anything else.
So I'm thinking of sending her a message where I basically say that I want to hang out with her, but I'm afraid it'll come across as too sudden and almost creepy which would turn her down.
What do you think ?