  We demonstrate a fast three-axis optical magnetometer using cold,
optically-trapped $^{87}$Rb gas as a sensor. By near-resonant Faraday rotation
we record the free-induction decay following optical pumping along two
orthogonal axes to obtain the three field components and one gradient
component. A single measurement achieves shot-noise limited sub-nT sensitivity
in 1 ms, with transverse spatial resolution of $\sim$ 20 $\mu$m. We make a
detailed analysis of the shot-noise-limited sensitivity.
