I am studying differential geometry and I'm still working fully understanding what a differential form is. I have consulted numerous textbooks and posts on this website, and what I find most often are the following two definitions of a $k$-form. For simplicity let's frame all this in $\mathbb{R}^n$.
A $k$-form is a function $\omega$ that assigns to each point $p \in U \subset \mathbb{R}^n$ a $k$-linear alternating function $\omega_p$ that is defined on the tangent space $T_p \mathbb{R}^n$. In other words, a differential $k$-form $\omega$ is the following function: $$\omega: U \rightarrow A_k(T_p \mathbb{R}^n)\\ p \mapsto \omega_p, \quad \omega_p: T_p \mathbb{R}^n \rightarrow \mathbb{R}.$$
A $k$-form $\omega$ is an alternating function such that $$\omega: T_p\mathbb{R}^n \rightarrow \mathbb{R}.$$
Let me point out a few differences. The first definition is specific to a point $p$, the second is not. The first definition takes a $k$-form to be a function that assigns another function (that is alternating and maps to a scalar field) to each point $p$, in contrast the second definition directly defines a $k$-form to be this $k$-linear alternating functional.
A few posts on here offer a geometric picture behind differential forms, such as $dx$ being the projection onto the $x$ coordinate, or that if $n$ vectors are fed into a $n$-form, it will spit out some scaled version of the volume of the parallelepiped in $n$-dimensions. These all suggest that definition 2 is correct, but it is definition 1 that I see more often in textbooks.
Which one is correct?