I as reading the book ELEMENTARY DIFFERENTIAL GEOMETRY, by O'NEILL, and it's definition of $m$-forms (differential forms) are about functions $\psi:U\subset\mathbb{R}^n\to \mathbb{R}$ generated by the product of $dx_i,\:i\in I$, where $I$ is a $m$-list of $\{1,\cdots, n\}$.
I had a professor that defined alternate forms by this way (in fact, he use only properties as alternation rules, etc, to define the forms, so he proves that this definitions leads to above definition). However, he defines differential $m$-forms as functions $\omega:U\subset\mathbb{R}^n\to \wedge^m(\mathbb{R}^n)$, ie, maps vectors (or points) into alternate forms.
I'd like to know if in some sense the two definitions can be similar.
More than that, in some point he wrote this example of $1$-form:
$$\omega:p\mapsto \omega(p)=a_1(p)du_1(p)+a_2(p)du_2(p)$$
Following the definition he gaves, I expect that $\omega (p)$ is an alternate form of degree $1$, ie, a map $U\subset\mathbb{R}^n\to \mathbb{R}$.
However, I understood $a_1(p)du_1(p)+a_2(p)du_2(p)$ as a point, not a map.
So, I am very confused with this definitions.
Many thanks in advance for any clarification.