For a chart $x:U\to\mathbb{R}^n$ on a pseudo-Riemannian manifold $(M,g)$, the volume form can be expressed as $$ \mathrm{vol}_g = \sqrt{|\det(g_{\cdot\cdot})|}\ \ \mathrm{d}x^1\wedge\cdots\wedge\mathrm{d}x^n $$ where $\det(g_{\cdot\cdot})$ is the determinant of the matrix representation of $g$ in the coordinates $x^i$. Calculation-wise, this works splendidly. However, formalism tells us that we can only take the determinant of endomorphisms (i.e. linear transformations) which are $(1,1)$-tensors. What I wonder is this: what is the endomorphism $\phi_{g,x}$ that is truly being determined in $$ \det(\phi_{g,x}) = \det(g_{\cdot\cdot}) $$ I know that $\phi_{g,x}$ depends on $x$ because the matrix representation of $g$ under coordinates $x^i$ produces the extra factor needed to keep the volume form invariant, but I do not know how this map can be defined.
Question: What is the endomorphism $\phi_{g,x}$ that satisfies $\phi^a_b=g_{ab}$ and thus $\det(\phi) = \det(g_{\cdot\cdot})$ for coordinates $x^i:U\to\mathbb{R}$?
Let $V$ be an $n$-dimensional vector space over $\Bbb{R}$, let $g:V\times V\to\Bbb{R}$ be a non-degenerate symmetric bilinear form and let $\beta$ be a basis of $V$. Then, relative to this basis, we can calculate the matrix representation $[g]_{\beta}$, and this has a certain determinant. To do this "abstractly", what we need to do is consider the basis induced isomorphism $\psi_{\beta}:V\to V^*$ (i.e the one which maps the basis $\beta$ onto the dual basis $\beta^*$), and consider the "flat map" or the "musical isomorphism" $g^{\flat}:V\to V^*$ defined by $g^{\flat}(v):=g(v,\cdot)$.
Then, consider the composition $\phi_{g,\beta}:= \psi_{\beta}^{-1}\circ g^{\flat}:V\to V$. This is the right thing to consider as we easily see that the matrix representation of this endomorphism relative to the basis $\beta$ (we use this basis on the domain and target) is exactly what we'd like: $[\phi_{g,\beta}]_{\beta}^{\beta}= [g]_{\beta}$. Therefore, \begin{align} \det(\phi_{g,\beta})=\det\left([g]_{\beta}\right), \end{align} where the determinant on the left is the "abstract" one defined for any endomorphism.
In the manifold setting, given a chart $(U,x)$, we consider the chart-induced basis $\left\{\frac{\partial}{\partial x^i}\right\}_{i=1}^n$ and apply the above construction pointwise at every tangent space to get the desired $\phi_{g,x}$.