So I have tried to get motivation behind the formal definition of differential forms and that what I understood and I want to make sure that I’m on the right track:
So we want to integrate over a manifold which generalizes integration on $\mathbb{R}^n$, so we want to talk about something that looks similar enough to $\mathbb{R}^k$ locally, so we define a manifold and then we want to use our knowledge of $\mathbb{R}^k$ and look at the manifold locally and define the tangent space to each point(and now we look at a space that is isomorphic to $\mathbb{R}^k$ and way easier to use). But now we have a new problem, we have a different coordinate system to each tangent space and we want to get rid of it so we define $dx_1,dx_2,\ldots,dx_k$ to be the basis (which will be the partial derivatives to each point $p$) for each tangent space and then we want to integrate over the manifold we want a way to define volume independent of coordinates so we will use the determinant which is an alternating multilinear function and it will approximate the $k$ dim volume at the point $p$ we chose for the tangent space.
So we basically got that a differential form basically gives to each point $p$ on a set defined on a $k$ dim manifold in $\mathbb{R}^{n}$ a volume function which is defined on the tangent space. And when we talk about $1$ form for example so each differential form will have the form of a linear combination of $$a_1(p)dx_1+a_2(p)dx_2+\ldots+a_n(p)dx_n$$ because each of the coefficients is dependent of $p$ we will get that each of them is a function of $p$, and because we the $1$ dim volume is length we got that $1$ form is measuring a length over a set in $1$ dim manifold on $\mathbb{R}^n$ so when we will integrate we will get the line integral. And in the same analogy we can have the surface integral but now we will measure an area over a set on a manifold.
Is my intuition correct? Or am I really far behind?
