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?