I was wondering if there is an easy argument that manifolds that are orientable always have a volume form and vice versa?
So I am not looking for a full proof of this, but rather a good argument how the normal vector field and the volume form relate to each other.
Especially, this should be simpler in dimension $2$ and I even got an idea how to construct the volume form, i.e. if $n$ is a normal field then
$$\omega( \xi, \eta) := \langle n , \xi \times \eta \rangle$$ defines a volume form.
Now the question would be, how does a volume form $\omega$ define a normal vector field?