I'm trying to understand the definition of the integral of a general $k$-form $\omega$ over a parameterised $k$-manifold $M$ :$$\intop_{M}\omega=\intop_{D}\omega\left(\frac{\partial\mathbf{X}}{\partial u_{1}},\ldots,\frac{\partial\mathbf{X}}{\partial u_{k}}\right)du_{1}\cdots du_{k},$$
where $D$ is a region of $u_{1},\ldots,u_{k}$ space. So I'm looking at the 2-form version of this integral for a parameterised surface $S$:$$\intop_{S}\omega=\intop_{D}\omega\left(\frac{\partial\mathbf{X}}{\partial u},\frac{\partial\mathbf{X}}{\partial v}\right)du\,dv.$$
I've come up with an intuitive “explanation” for this second integral (which also works for higher dimensions) that, at my elementary level appears convincing, but I may be wrong.
So my question is, is the following argument correct/valid?
First, I note that a 2-form $\omega=dx\wedge dy$ acting on two tangent vectors to $S$ is given by$$\left(dx\wedge dy\right)\left(\frac{\partial\mathbf{X}}{\partial u},\frac{\partial\mathbf{X}}{\partial v}\right)=\left|\begin{array}{cc} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v}\\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{array}\right|.$$
Then, from the change of variable formula, I see that$$dx\wedge dy=\left|\begin{array}{cc} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v}\\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{array}\right|\,du\wedge dv.$$
By substitution I then say$$\omega=\omega\left(\frac{\partial\mathbf{X}}{\partial u},\frac{\partial\mathbf{X}}{\partial v}\right)\,du\wedge dv.$$And integrate this to get the original integral.
$$\intop_{S}\omega=\intop_{D}\omega\left(\frac{\partial\mathbf{X}}{\partial u},\frac{\partial\mathbf{X}}{\partial v}\right)du\,dv.$$
This is my attempt to improve a previous question, which has not been well received.