Discrete electromagnetism (DEM)-in the authors' view-should be a self-consistent theory, mirroring the properties of the continuous electromagnetic theory in a discrete setting. Any recursion to continuous techniques can be interpreted as an inconsistency in the discrete theory. Recently, discrete Hodge operators on tetrahedra and triangles have been introduced that avoid the concepts of interpolation and integration of fields. In this paper we introduce a geometrical definition of a discrete Hodge operator for general dimension n and degree p,0lesnles3,0lesplesn. The definition generalizes previously published definitions. The increased level of abstraction allows for a short definition and a concise discussion of the properties of this operator