differential forms on a manifold

Hi guys. So I'm a little stuck on the concept of forms on a manifold. I understand what forms are, and understand their properties in R2 and R3 (exactness, closedness, exterior derivative of a form, etc) but I'm confused about how one finds forms on a manifold, and what it means for a form to be exact on a surface/manifold. I'm guessing it has something to do with the charts of that manifold? Say, for a 2-manifold M, would you just use the chart conversion for coordinates from R2->M, and this would give you a form on your manifold M?

Eg. A 1-form on the torus. a 2-form on the 3-sphere etc. Could someone give me a few examples of forms on typical manifolds like these and give a bit of an explanation..

Thanks!