I am trying to understand how to construct an integral is a basic algebraic manner using the following tools. Let $\{ \hat{\mathbf{x}}_1, \dots, \hat{\mathbf{x}}_n\}$ be an orthonormal basis such that $\hat{\mathbf{x}}_i \cdot \hat{\mathbf{x}}_j=\delta^i_j$. Let us define a curvilinear basis:
$$ \mathbf{e}_i=\sum_{k=1}^n a_i^k \hat{\mathbf{x}}_i $$
Then this relation holds (ref):
$$ \mathbf{e}_1\wedge\dots \wedge \mathbf{e}_n=\sqrt{g} (\hat{\mathbf{x}}_1\wedge \dots\wedge\hat{\mathbf{x}}_n) \tag{1} $$
We note that this relation has a similar form as the Riemannian volume form:
$$ \omega=\sqrt{g} dx_1\wedge \dots \wedge dx_n \tag{2} $$
My goal now is to go from (1) to (2).
If I take $\hat{\mathbf{x}}_i$ to be a vector part of the tangent space of some manifold $M$, I can make this replacement:
$$ \hat{\mathbf{x}}_i=\frac{\partial }{\partial x_i} $$
which gives me the following:
$$ \begin{align} \mathbf{e}_1 \wedge \dots \wedge \mathbf{e}_n &= (\sum_{k=1}^n a_i^k \frac{\partial}{\partial x_k})\wedge \dots \wedge (\sum_{j=1}^n a_i^j \frac{\partial}{\partial x_j})\\ &=\sqrt{g} (\frac{\partial}{\partial x_1} \wedge \dots \wedge \frac{\partial}{\partial x_n} ) \end{align} $$
But I do not understand how I can go from $\partial / \partial x_i$ to $dx_i$?