Does it make sense to add two differential forms with different degrees like $dx+dx\bigwedge dy$? If yes, what's the arguments of it?
I ask this because in text book, the vector space, $\Omega^*(M)$, of $C^\infty$ differential forms on a manifold $M$ is defined as
$\Omega^*(M)=\bigoplus_{k=0}^n\Omega^k(M)$,
where $\Omega^k(M)$ is the vector space of k-forms. It means each element in $\Omega^*(M)$ is uniquely a sum $\sum_{k=0}^n \omega_k$, $\omega_k\in \Omega^k(M)$.