  We derive a closed analytical expression for the exchange energy of the
three-dimensional interacting electron gas in strong magnetic fields, which
goes beyond the quantum limit (L=0) by explicitly including the effect of the
second, L=1, Landau level and arbitrary spin polarization. The inclusion of the
L=1 level brings the fields to which the formula applies closer to the
laboratory range, as compared to previous expressions, valid only for L=0 and
complete spin polarization. We identify, and explain, two distinct regimes,
separated by a critical density $n_c$. Below $n_c$, the per-particle exchange
energy is lowered by the contribution of L=1, whereas above $n_c$ it is
increased. As special cases of our general equation we recover various known,
more limited, results for higher fields, and identify and correct a few
inconsistencies in some of these earlier expressions.
