@Thorgott Here's the relevant blackboxes. Given any Riemannian manifold $(M, g)$, for any point $p \in M$ $\DeclareMathOperator{\vol}{vol}$
(1) There's $r > 0$ and a diffeomorphism $\exp_p : B(0, r) \subset T_p M \to M$ such that $d\exp_p(0) = I$. The image is called geodesic ball of radius $r$ around $p$. For a reference, check do Carmo.
(2) In local coordinates $(x_1, \cdots, x_n)$ given by $\exp_p$ around $p$, the Riemannian volume form is related to the form $dx_1 \wedge \cdots \wedge dx_n$ by $d\vol_g = f(x) dx_1 \wedge \cdots \wedge dx_n$ where $f(x) = 1 + O(\|x\|^2)$. This follows f…