  We investigate theoretically the phase diagram of a spin-orbit coupled Bose
gas in two-dimensional harmonic traps. We show that discrete Landau levels
develop at strong spin-orbit coupling. For a weakly interacting gas, quantum
states with skyrmion lattice patterns emerge spontaneously and preserve either
parity symmetry or combined parity-time-reversal symmetry. These phases can be
readily observed by experimentally engineering spin-orbit coupling and
interatomic interactions for a cloud of $^{87}$Rb atoms in a highly oblate
trap.
