Generally speaking after several romantic dates it's the woman that starts to ask the man questions such as "What are we?" or "Where is this going?" :-)
Seriously if you have had multiple dates playfully flirting with one another, laughing, holding hands, ending with good-night kisses, talk to each other on the phone nightly and so forth over a few weeks it should be easy to tell her you want an "exclusive" relationship.
It would be a mistake to ask her to be your "girlfriend" or be "exclusive" if she has not had the opportunity to have some dates with you to see if there is any chemistry between you. To do otherwise would be equivalent to asking someone to buy a car they have never driven or ridden in. She has to date you a few times before she can decide if she wants to see you exclusively.