  The classical Rainich(-Misner-Wheeler) theory gives necessary and sufficient
conditions on an energy-momentum tensor $T$ to be that of a Maxwell field (a
2-form) in four dimensions. Via Einstein's equations these conditions can be
expressed in terms of the Ricci tensor, thus providing conditions on a
spacetime geometry for it to be an Einstein-Maxwell spacetime. One of the
conditions is that $T^2$ is proportional to the metric, and it has previously
been shown in arbitrary dimension that any tensor satisfying this condition is
a superenergy tensor of a simple $p$-form. Here we examine algebraic Rainich
conditions for general $p$-forms in higher dimensions and their relations to
identities by antisymmetrisation. Using antisymmetrisation techniques we find
new identities for superenergy tensors of these general (non-simple) forms, and
we also prove in some cases the converse; that the identities are sufficient to
determine the form. As an example we obtain the complete generalisation of the
classical Rainich theory to five dimensions.
