Let $M$ be a smooth manifold and $E \to M$ a vector bundle over $M$ with a connection $\nabla$. Locally on an open set $U \subset M$ with a frame $(E_1,\dots,E_k)$, we can write any section $s$ of $E|_U$ as $s = \sum c^iE_i$ for smooth functions $c^i:U \to \Bbb R$. This means that we can write
$$ \begin{align*} \nabla_X s &= \nabla_X\left(\sum_i c^i E_i\right) \\ &= \sum_i \nabla_X(c^iE_i) \\ &= \sum_i (Xc^i)E_i+c^i\nabla_X E_i \end{align*} $$
or alternatively
$$ \begin{align*} \nabla s &= \nabla\left(\sum_i c^i E_i\right) \\ &= \sum_i \nabla(c^iE_i) \\ &= \sum_i dc^i \otimes E_i+c^i\nabla E_i. \end{align*} $$
In any case this can be computed by knowing $\nabla E_i$'s. Since $\nabla E_i$ is a section of $E$ over $U$ it is a linear combination of $E_j$'s and so
$$ \nabla_X E_i = \sum_j \omega^j_i(X)E_j $$
and we define the connection $1$-forms $\omega = [\omega^j_i]$ to be the coefficients in this sum.
Some authors state that locally any connection can be locally written as $$d + A$$ where $A$ is a matrix valued one-form. Could anyone here elaborate on how this conclusion is made, and is it possible to derive it from the matrix of connection $1$-forms as these two would seem like they are closely related. I have a suspicion that the latter uses the isomorphism $E|_{U} \cong U \times \Bbb R^k$ in some way, but not sure how.
Fix any connection $\nabla_0$ on $E$, and let $\nabla$ be an arbitrary connection on $E$. Consider the operator $\nabla_0-\nabla:\Gamma(E)\to\Gamma(E\otimes T^*M)$. You can show that this is an endomorphism valued $1$-form: $$(\nabla_0-\nabla)(fs)=df\otimes s +f\nabla_0s-df\otimes s -f\nabla s =f(\nabla_0-\nabla)s$$ This $C^\infty(M)$-linearity is what characterises $\nabla_0-\nabla$ as a tensor - in this case a $1$-form. Thus, since the difference of any two connections is an endomorphism valued $1$-form, it is also true that any connection can be obtained from a fixed basepoint $\nabla_0$ by adding an endomorphism valued $1$-form. In particular, on a trivialisation, one can take $d$ to be that basepoint, and then locally any connection can be expressed as $\nabla=d+A$. So yes, the local isomorphism $E|_U\cong U\times\mathbb{R}^r$ is crucial, because it gives a basepoint for the space of connections (in the given trivialisation, of course).