The goal of this paper is to extend Loday's Leibniz cohomology from a Lie algebra invariant to an invariant for differentiable manifolds so that Leibniz cohomology is a non-commutative version of de Rham cohomology. The non-commutativity arises by considering a cochain complex of tensors (from differential geometry) which are not necessarily skew-symmetric. Leibniz cohomology, HL^*, is no longer a homotopy invariant---in fact the first obstruction to the homotopy invariance of HL^*(R^n) is the universal Godbillon-Vey invariant in dimension 2n+1. The main calculational result of the paper is then a calculation of HL^*(R^n) in terms of (i) certain universal invariants of foliations, and (ii) Loday's product structure on HL^*. Unlike Lie algebra or Gelfand-Fuks cohomology, the Leibniz cohomology of vector fields on R^n contains infinite families of elements which support non-trivial products.