Giving a girl in college a note? Is it a good idea?

So there is this woman who I met at the beginning of this year. We are both into each other but when I asked her out, she told me she had a boyfriend. I simply took it as a rejection and as much as I was crushed and confused (She is the one who initiated all the eye contact so that's why I was left like huh?) I tried to move on.

To add to the confusion, she continued to make eye contact with me through the rest of the semester and I never talked to her again.

Now its fall and I still see her everyday. I think she is single now and I want to settle my curiosity once an for all. I was thinking of going up, make a little friendly chat and then when I'm about to leave give her a little note saying something along the lines of

I shoulve done this a long time ago but for some reason you have been on my mind since we last talked and maybe its just me but I feel like we should know each other.

I'll add more but that's going to be the main message I'm trying to get across. Do you think it's a good idea?
Giving a girl in college a note? Is it a good idea?
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