Can you find a 2-form not written as the wedge of two 1-forms?

Question

I was under the impression that all 2-forms are the wedge $(\wedge)$ of two 1-forms. Is it possible to have a 2-form that you can't write as $A\wedge B$ with $A,B$ 1-forms?

Answer 1

Yes, it is possible. (And you should find an example yourself: I will not deprive you of the joy of finding it :) )