A new formulation for treating spinor equations on a spacetime is introduced and applied to the spin-2 equation for the Weyl spinor in vacuum general relativity. The power of the formalism rests on the fact that it is index free, describing structures in terms of algebraic relations which makes it well adapted for use in algebraic manipulation programs. The starting point is the fact that connections in bundles can be viewed as derivations on the algebra of sections of the bundle. In the case of the spin bundle there is a canonical operator basis that allows one to take components in a canonical way so that one can express everything in terms of scalar operators. Thus, essentially, this is a non-commutative Newman-Penrose formalism. In an application to general relativity we present an algorithm that recursively produces the terms of a Taylor series expansion of the Weyl spinor around the apex of a light cone from characteristic data given on that cone.