During the course, we have defined differential forms as maps $$ \omega : D(M) \times \cdots \times D(M) \rightarrow \mathcal{C}^\infty(M), $$ where $D(M)$ are the global derivates of the manifold $M$. I know that locally, a global derivate has the form $$ \sum_{i=1}^n a^i \partial/\partial x^i, $$ where $x^1,...,x^n$ are the local coordinates and $a^i$ are $\mathcal{C^\infty}$ functions.
I don't understand why an antisymmetric $k$-differential form has this form in a neighbouhood $U$: $$ \sum a_{i_1,...,i_k}dx^{i_1} \wedge \cdots \wedge dx^{i_k}. $$ Where $dx^i$ forms a basis of the covector space $T_pM^*$.
Said this... Locally, I understand that $\omega$ depends on the $\partial/\partial x^i$, so that the $a_{i_1,...,i_k}$ are determined but the action of omega on the $\partial/\partial x^i$, but I don't understand where the wedge product comes from.
If someone could illuminate me, I'd be thankful!