You need to write $\omega X$ as $\omega \otimes X$.
If you prove that $R: \Gamma(E) \to \Omega^2(M,E)$ is tensorial - i.e. $R(X,Y)(fS) = fR(X,Y)S$ where $S\in\Gamma(E)$, $f\in\mathcal{C}^\infty(M)$ and $X,Y\in TM$, then it follows that this mapping is indeed an element of $\Omega^2(M,End(TM))$. (That is even more elementary result - see any introductory book on differential geometry.)
The classical formula for curvature follows directly from the definition of the action of $\nabla$ on $\Omega^p(M,E)$.