In this paper, we consider the relative equations of spacecraft motion in terms of a new set of orbital elements that includes the components of the angular momentum in the inertial reference frame, the components of the eccentricity vector in a suitably chosen non-inertial reference frame, and a true-anomaly-like angle determining the location of the spacecraft. We derive a detailed mathematical formulation of the relative dynamics using orbital element differences in terms of the new state parameters. Using numerical simulations, we compare the formulation with Hill-Clohessey-Wiltshire equations of relative motion.