How can I justify the definition of differential form on manifolds? Algebraically I'm ok with it but I'm not very geometrically motivated. I mean this definition:

> Given a differentiable manifold X, or even a generalized smooth space X for which this definition makes sense, a differential form on X is a section of the exterior algebra of the cotangent bundle over X.

I can understand what this is saying, but I can't understand why this is the case. Is there anything thoughtful about being a "section" here (maybe Tao's word can ring a bell, but I haven't found it yet)?

I'm ok with exterior algebrab of cotangent bundle.

* We care about cotangent spaces because they are dual to tangent spaces. A tangent space is spanned by differential operators (partial over partial x). A cotangent space, being dual to the corresponding tangent space, is spanned by dx, dy,... This amounts to integration.

* We need wedge products because we care about the order (signed), and wedge product gives rise to determinant, which means signed, oriented volume.

Correct me if I'm wrong but I think I get it right. But still, I find no way to reason being a section. Where should I set my mind to?