Let $M$ be a smooth oriented $k$-manifold on $\mathbb{R}^n$, with a parametrization $g\colon U \to \mathbb{R}^n$ where $U \subseteq \mathbb{R}^k$. Suppose we have a continuous function $f\colon \mathbb{R}^n \to \mathbb{R}$ with compact support, and we wish to integrate it over $M$. We want then to calculate:
$$\int_M f dV$$
Where $dV$ is a volume element, a differential $k$-form particular to this manifold $M$. I've read on wikipedia that the pullback of the volume element is given by:
$$g^*(dV) = \left|g'^T \cdot g'\right|dx_1 \wedge\cdots \wedge dx_k$$
Where $dx_i$ are the usual projection forms in $\mathbb{R}^k$ and $g'$ is the Jacobian matrix of $g$. I would like to understand why though. I'm not even sure what a formal definition of the volume element of a $k$-manifold on $\mathbb{R}^n$ is.
Sure, if $k = n$, then this will simply be $dV = dx_1\wedge \cdots \wedge dx_n$ with the projections happening in $\mathbb{R}^n$. The pullback is easy to calculate then, as it commutes with exterior products and exterior derivatives. Outside this realm, I'm quite lost on what's going on. What is the length element of a curve in $\mathbb{R}^2$? Or the area element of a surface in $\mathbb{R}^3$? How does all of this generalizes to $k$-manifolds over the $n$-space?