Explicit generators for high order ($r>1$) scalar and vector finite element spaces generally used in numerical electromagnetism are presented and classical degrees of freedom, the so-called moments, revisited. Properties of these generators on simplicial meshes are investigated and a general technique to restore duality between moments and generators is proposed. Algebraic and exponential optimal $h$- and $r$-error rates are numerically validated for high order edge elements on the problem of Maxwell's eigenvalues in a square domain.