Starting with the curvature 2-form a recursive construction of totally antisymmetrised 2p-forms is introduced, to which we refer as p-Riemann tensors. Contraction of indices permits a corresponding generalisation of the Ricci tensor. Static, spherically symmetric ``$p$-Ricci flat'' Schwarzschild like metrics are constructed in this context for d>2p+1, d being the spacetime dimension. The existence of de Sitter type solutions is pointed out. Our 2p-forms vanish for $d<2p$ and the limiting cases d=2p and d=2p+1 exhibit special features which are discussed briefly. It is shown that for d=4p our class of solutions correspond to double-selfdual Riemann 2p-form (or p-Riemann tensor). Topological aspects of such generalised gravitational instantons and those of associated (through spin connections) generalised Yang-Mills instantons are briefly mentioned. The possibility of a study of surface deformations at the horizons of our class of ``p-black holes'' leading to Virasoro algebras with a p-dependent hierarchy of central charges is commented on. Remarks in conclusion indicate directions for further study and situate our formalism in a broader context.