  This note contributes to the structure theory of abstract rigidity matroids
in general dimension. In the spirit of classical matroid theory, we prove
several cryptomorphic characterizations of abstract rigidity matroids (in terms
of circuits, cocircuits, bases, hyperplanes). Moreover, the study of
hyperplanes in abstract rigidity matroids leads us to state (and support with
significant evidence) a conjecture about characterizing the class of abstract
rigidity matroids by means of certain "prescribed substructures". We then prove
a recursive version of this conjecture.
