Denote $G:=Hol(g) \subset SO(n)$ and $\mathfrak{g}:=\mathfrak{hol}{g} \subset \mathfrak{so}(n)$. The Levi-Civita connection $A \in \Omega^1(GL(M), \mathfrak{gl}(n))$ reduces to a connection $\hat{A} \in \Omega^1(G(M), \mathfrak{g})$.
The curvatures of $A$ and $\hat{A}$ are 2-forms in the following spaces: $F_A \in \Omega^2(GL(M),\mathfrak{gl}(n))$, $F_\hat{A} \in \Omega^2(G(M), \mathfrak{g})$. Alternatively, we can view $F_A$ (or $F_\hat{A}$) as a curvature endomorphism on the associated fiber bundle $GL(M) \times \mathbb{R}^n \simeq G(M) \times \mathbb{R}^n \simeq TM$. I.e. $R \in \Gamma ( \Lambda^2T^*M \otimes End(TM,TM))$.
Now, in what way is $\mathfrak{g} \subset End(TM,TM))$ and why does $\nabla$ take values in $\mathfrak{g}$?
It does not seem too far fetched, because $\hat{A}$ takes values in $\mathfrak{g}$, but I do not know how to make this precise.