Problem:
Show that a $k$-form $\omega$ is smooth if only if it is smooth as a map $\omega \colon M\rightarrow \Lambda ^k(M)$.
Help for a clearer solution this problem. I have already seen that this issue was posted in this post: Help to condition: "smooth as a section $\omega :M\rightarrow \Lambda ^k(M)$ of $\pi$"
but I wanted a somewhat clearer solution. To show that $\omega$ is a smooth k-form I'm trying to use this characterization:
$\omega$ is a $k$-form smooth of $M^n$ if only if $$\omega(X_1,\dots,X_k):M\rightarrow \mathbb{R}, \text{ give by } \omega(X_1,\dots,X_k)(p)=\omega_p((X_1)_p,\dots,(X_k)_p)(p)$$ is smooth function for every collection $X_1,\dots,X_k$ of smooth vector fields on $M$.
Remark: Consider $\Lambda ^k(M)=\bigcup_{p\in M}\Lambda ^k(T_{p}M)$ with the natural smooth structure, $M$ a smooth $n$-manifold. With this structure on $\Lambda ^k(M)$, the $\pi \colon \Lambda ^k(M)\rightarrow M$ projection is smooth.