The effect of a weak Zeeman spin splitting on the spatial distribution and the energy of an isolated skyrmion in a 2D electron gas under a strong magnetic field at filling factor is studied within the framework of the Hartree-Fock approximation. Finite Zeeman splitting introduces two different characteristic lengths into the problem, corresponding to the tail and the core of the spin distribution. The non-linear differential equation for the macroscopic spin density associated with the skyrmion is solved in the limit of very small g-factor, for which the tail of the skyrmion is much longer than its core radius. In this limit the Coulomb repulsion energy of the skyrmion, which determines the core radius, does not affect the shape of the spin distribution.