We represent the ensemble of extremal copositive matrices A ∈ COP^n of the copositive cone as a disjoint union of a finite number of algebraic manifolds. These manifolds are strata of algebraic sets, each of which is characterized by a finite collection E of pairs of index sets. We give necessary conditions on E to correspond to an algebraic set containing exceptional extremal matrices of COP^n. A manifold of extremal matrices is essential if it is not contained in the closure of another such manifold. We compute the essential manifolds of extremal matrices of the cone COP^6.