The cyclotron spin-flip excitations of a two-dimensional electron system in the extreme quantum limit have been investigated by inelastic light scattering. The electron interaction energy was directly determined from the gap between the charge- and spin-density modes in a broad region of electron filling factors. A comparison of the experimental results with the Hartree-Fock and generalized single mode approximations indicates a strong correlation between electrons at filling factors corresponding to both compressible and incompressible states.