Was going to put this as a comment on Willie's answer, but it was getting pretty long:

The idea is that, while a connection is *not* a section of $\Omega^1(M,End(E))$- one may view it as a map $\Gamma(E)\to \Gamma(T^*M \otimes E)$- that is, if you give a connection a section, it will give you a peculiar creature which eats up vector fields and spits out **what look like sections of E** as the $T^*M$ tensor factors go to scalars. 

In reality these things that look like sections are morally our connection's choice of lift of the vector field to E (- the confusing factor being that because E is a vector space, it is its own tangent space). Now, as already mentioned above a connection is globally $\mathbb{C}$-linear, but not locally (that is not $C^\infty(M)$-linear) as would be reqired to have $\nabla \in \Gamma(\Omega^1(M,End(E))) $. 

This is because any candidate for a lift must perform the dirty deed of differentiating the section along your vector field (this is what the Leibniz rule condition is about). With this in mind, our first draft of a connection would just be $\Theta=$ an extended $d_{DeRham}$-taking sections to their derivatives tensored with the duals of the directions in which they are differentiated (a so called 'trivial' connection)- but there is still room for manoevre and we can add to $\Theta$ a 1-form $A \in \Gamma(\Omega^1(M,End(E)))$ with coefficients in $End(E)$ and still get something that satisfies our conditions.

In fact for any connection $\nabla$ we may write $\nabla= \Theta +A$ for some $End(E)$ valued 1-form A. Now in $\nabla \circ \nabla$ 'the $\Theta \circ \Theta $ bit' will go straight to zero for the same reason $d^2=0$ in the DeRham complex and you will have yourself just an $End(E)$ valued 2-form left (again, strictly, an $End T_{\Gamma (x)}E$ valued 2-form, but who's counting?)- as you said, the curvature $R \in \Omega ^2 (M,E)$.

So (and this is the actual content of what I was going to put in the comment- the rest was just me getting carried away) what is an $End(E)$ valued 2-form when it's at home? 

It's something that takes in pairs of vector fields and spits out an element of $End(E)$

$\iff$ It's something that takes in pairs of vector fields and spits out a $Rank(E) \times Rank(E)$ matrix 

$\iff$ It's something that takes in pairs of vector fields and spits out the entries of a $Rank(E) \times Rank(E)$ matrix 

$\iff$ it is a $Rank(E) \times Rank(E)$ matrix with entries in $\Omega^2 (M)$

So we can add the identity without fear, and we can sensibly take the determinant since even forms commute when we multiply them.