Quote by pervect
I think part of the machinery that makes field lines work is that you can write a general twoform as the sum of two wedge products of oneforms in 4 dimensions, and this is peculiair to 4d.

This has nothing to do with it. Field lines work because the sourcefree Maxwell equations imply that the 2form F is
harmonic.
Harmonic forms (in any dimension) have the distinction that they capture purely topological information. If you integrate a harmonic nform over a closed nsurface, the result is either zero or nonzero, depending on whether the nsurface encloses some topological feature (for example, a 1surface on a cylinder might wrap around the cylinder...or a 2surface in R^3 might enclose a charge).
Any nsurface that encloses the same set of topological features must give you the same result.
You can think of this as a higherdimensional analogue of contour integration. In fact, all analytic functions on the complex plane satisfy Laplace's equation, which is why contour integration works.