Let $E\to M$ be a $C^\infty$ vector bundle with a connection $\nabla$. Let $A^{k}(M;E) $ denote the vector space of $C^\infty$ $k$-forms with values in $E$. I am trying to define the covariant exterior differential, $D:A^{k}(M;E)\to A^{k+1}(M;E)$, uniquely characterized by the formula $D(\omega \otimes s)=d\omega \otimes s +(-1)^k \omega \nabla s$ for $s\in \Gamma(E)$ and $\omega \in A^{k+1}(M)$. Here $\Gamma(E)$ is the vector space of all $C^\infty $ sections of $E\to M$, and $A^{k}(M)$ is the vector space of all $C^\infty$ $k$-forms on $M$.
To do this, I should show that every $\tau\in A^{k}(M;E)$ can be written as a finite linear combination of sections of the form $\omega \otimes s$. I know this is locally true, but is this also true in general?