I came across this following way of defining connection and curvature which is not so obviously equivalent to the definitions as familiar in Riemannian Geometry books like say by Jost.

If $E$ is a complex vector bundle over a manifold $M$ then one defines the space of vector valued $p$-differential forms on them as $\Omega^p(M,E) = \Gamma ( \wedge ^p (T^*M) \otimes E) $

The connection be defined as the map , $\nabla: \Gamma(E) \rightarrow \Omega^1(M,E)$ satisfying $\nabla(fX) = (df)X+f\nabla X$ where $f \in C^{\infty}(M)$ and $X \in \Gamma(E)$

{It might help if someone can make explicit as to what exactly is $(df)X$"? To make sense this has to be an element of $\Omega^1(M,E)$}

Now one defines curvature of the connection as the map, $$R = \nabla \circ \nabla : \Gamma(E) \rightarrow \Omega^2(M,E)$$

Now it is not clear to me as to how this makes $ R \in \Omega^2(M,End(E))$ ?

And how does this match up with the more familiar form as,

$R(X,Y) = \nabla _X \nabla _Y - \nabla _Y \nabla _X - \nabla _{[X,Y]}$

One can extend the definition of connection to be a map $\nabla: \Omega^p(M,E) \rightarrow \Omega^{p+1}(M,E)$ such that for $\omega \in \Omega^p(M)$ and $X \in \Gamma(E)$ one has,

$$\nabla(\omega X) = (d\omega)X + (-1)^{deg \omega}\omega \wedge \nabla X$$

{Here again it would be helpful if someone can can explain what is the meaning of $(d\omega)X$"? To make sense it has to be an element of $\Omega^{p+1}(M,E)$}

This looks very much like the de-Rham derivative of vector valued differential forms but the notation looks uneasy.