Let $E\to X$ is a a (smooth real) vector bundle with structure group some Lie group $G$. Suppose we have a (linear) connection $\nabla$ on $E$.

Is it true that if $A$ is the connection 1-form of the connection $\nabla$ in local coordinates in some trivialisation, regarded as a 1-form with matrix coefficients, then this matrix is in $\mathfrak{g}$ the Lie Algebra of $G$?

When one defines a connection in a principal $G$-bundle one constructs the connection in this way. So it would make sense that if $E$ has structure group $G$ , then this happens too.

However, I have never seen this argument when working with vector bundles only. Furthermore, I have seen proofs trying to deduce properties of the connection $1$-form that would be trivial if we had that the matrices are in the Lie Algebra. So that makes me think that this is not true.

*The application I have in mind is the following:* If I have a real vector bundle of rank $k$ with structure group the orthogonal group $O(k)$ and $A$ is the connection $1$-form in local coordinates in some trivialisation. Would it always be the case that $A^\top=-A$?

*Note:* By structure group, I mean the sections of automorphisms defined on the fibres of $E$ that are induced by the transition functions between trivialising neighbourhoods of $E$.