$d+\Gamma$-understanding connections

Question

$d+\Gamma$ is a formula commonly found in textbooks when one reads about connection. To be more precise: if $U$ is an open set over which a given vector bundle $E \to M$ is trivial then a coonection $\nabla :C^{\infty}(M,E) \to \Omega^1(M,E)$ may be written, locally over $U$, in the form: $\nabla=d+\Gamma$. Here come my questions about these formula: I would like to ensure myself whether I understand it correctly. The space of all connections is an affine space: the difference between two connections is a homomorphism, to be more precise it is an element of $\Omega^1(M,End(E))$. When the bundle is trivial one can define at least one connection using the differential $d$. Therefore $d$ corresponds to the distinguished point in the affine space and $\Gamma$ is a vector of the underlying vector space. The action of $d$ is understood as follows: choose a local frame $(e_a)_a$: each section $s$ of $E$ may be written in the form $s=\sum_a s^ae_a$ and $ds:=\sum_a ds^a \otimes e_a$. On the other hand $\Gamma$ may be expressed as $\sum_i dx^i \otimes \Gamma_i$ where $\Gamma_i$'s are sections of $End(E)$. Therefore the action of $\Gamma$ on $s$ is the following:
$\Gamma(s)=\sum_{i,a,b}s^a \Gamma^b_{ia}dx^i \otimes e_b$ where $\Gamma_i=[\Gamma^b_{ia}]_{a,b}$ and $\Gamma^b_{ia} \in C^{\infty}(M)$. Please confirm whether it is ok or requires some corrections. I hope that maybe somebody will find this explanation useful.