  We develop a bosonization scheme for the two-dimensional electron gas in the
presence of an uniform magnetic field perpendicular to the two-dimensional
system when the filling factor \nu = 1. We show that the elementary neutral
excitations of this system, known as magnetic excitons, can be treated
approximately as bosons and we apply the method to the interacting system. We
show that the Hamiltonian of the fermionic system is mapped into an interacting
bosonic Hamiltonian, where the dispersion relation of the bosons agrees with
previous RPA calculations. The interaction term accounts for the formation of
two-boson bound states. We discuss a possible relation between these
excitations and the skyrmion-antiskyrmion pair, in analogy with the
ferromagnetic Heisenberg model. Finally, we analyze the semiclassical limit of
the interacting bosonic Hamiltonian and show that the results are in agreement
with those derived from the model of Sondhi {\it et al.} for the quantum Hall
skyrmion.
