Let w be a 3-form on a 4-manifold, and U be the set where w is nonzero. w is closed iff for all p in U there exists a neighborhood of p where w has coordinate representation 1 dx dy dz.
To show the reverse direction I just took d and got dw=0 so it's closed. I can't figure out the forward direction. I've tried to do it by linearity but that results in a system of equations I don't know how to solve. Any ideas?
If $dVol$ is a volume form, there exists a vector field $V$ such that $i(V) dVol=\omega$. Indeed if $\omega = a dXdYdZ+bdYdZdT+..$ and $dVol= dXdYdZdT$, take $X= a{\partial f\over \partial T}+b{\partial f\over \partial X}+..$
Then, as $V\not=0$, we can choose a new system of coordinates $(x,y,z,t)$ such that $V= {\partial\over \partial t}$, and in this coordinates system, $dVol=f(x,y,z,t)dxdydzdt$, and $\omega= f(x,y,z,t)dxdydz$.
Then, $d\omega=0$ means that ${\partial f\over \partial t}=0$. Therefore $\omega= f(x,y,z)dxdydz$, is basically a volume form on $\bf R^3$, and the standard "Moser lemma" produces a new system of coordinates where $\omega= dxdydz$