I have been trying to understand my situation before I move forward or do something stupid.
I met someone in the company I work (we don`t work together). It was attraction at first sight, then we started to know each other from work and we liked that. And after 3 months we went out.

We have been seeing each other once a week now for the last 3.5 weeks. Last time she told me she has a chat with her mom and told her she thinks that I am into her but not sure... I smiled. She told me the first time that she does not want to get trapped. But her whole family know about me and I even met the parents once when we went out. My girl is even starting to take contraceptive pills this week. The girl does not like to text or call as an ex-bf is stalking her and she does not like to check her phone anymore? I don't understand that.

Do I need to ask her to be my girlfriend or is that a game, it feels strange. En-light me!

I think you need to get these unsure things cleared up before you ask her to be your girl. If she doesn't like to text or call how do you communicate with her?
Learn more about her before you commit yourself to a relationship with her.

Tell her you are as uncertain about how she feels about you as she is uncertain what your feelings are toward her and you would like to know where you stand with her (or as old fashioned dad's use to ask ''what are your intentions with my daughter?)
''what are your intentions with me?''

It's only been a month, and there is much more to learn so don't trip over small things you don't understand yet. Just keep dating and having fun, and see what happens.