We attempt to define Seiberg–Witten equations on a closed, spin, oriented 4–manifold X, where a fixed spinor bundle s of X is twised by a U(n)–vector bundle E (n > 1). Like the abelian case of n = 1, we shall show hat the moduli space of the solutions to our Seiberg–Witten equations is also a finite dimensional compact manifold. We hope to extract some sort of topological invariant for spin 4–manifolds from investigating the higher rank monopoles of the U(n) Seiberg–Witten equations. We suspect that the higher rank invariants should not give us any new topological information. This is a work in progress.