  Using field effect devices with side gates, we modulate the 2 dimensional
electron gas hosted at the LaAlO$_3$/SrTiO$_3$ interface to study the
temperature and doping evolution of the magnetotransport. The analysis of the
data reveals different transport regimes depending on the interplay between the
different (elastic, inelastic, and spin-orbit) scattering times and their
temperature dependencies. We find that the spin-orbit interaction strongly
affects the low temperature transport in the normal state in a very large
region of the phase diagram, extending beyond the superconducting dome.
