I have a doubt that I have not seen answered/reflected anywhere, and is that: suppose that $M$ is a $k$-dimensional oriented regular submanifold of $\mathbb{R}^n$. If $k=n-1$ then there is a induced volume form (induced from the canonical Euclidean metric) what is given by $i_N\omega$, where $i_N$ is interior multiplication of the volume form $\omega$ in $\mathbb{R}^n$ by the unit normal outward-pointing vector field to $M$.
However if $k<n-1$ then $\dim T^\bot M>1$, so if I apply the interior multiplication various times using a basis of $T^\bot M$ I get a volume form to $M$. So this would be a way to give a canonically induced volume form to any regular and oriented submanifold of $\mathbb{R}^n$.
Is this approach correct, or is something else needed to make sense? If its correct, it is unique? I mean, as the unique way to define a volume form induced from the Euclidean metric?