  We develop an extended pump-probe Faraday rotation technique to study the
submicrosecond electron spin dynamics with picosecond time resolution in a wide
range of magnetic fields. The electron spin dephasing time $T_2^*$ and the
longitudinal spin relaxation time $T_1$, both approaching $250$ ns in weak
fields, are measured thereby in $n$-type bulk GaAs. By tailoring the pump pulse
train through increasing the contained number of pulses, the buildup of
resonant spin amplification is demonstrated for the electron spin polarization.
The spin precession amplitude in high magnetic fields applied in the Voigt
geometry shows a non-monotonic dynamics deviating strongly from a
mono-exponential decay and revealing slow beatings. The beatings indicate a two
spin component behavior with a $g$-factor difference of $\Delta g \sim
4\times10^{-4}$, much smaller than the $\Delta g$ expected for free and
donor-bound electrons. This $g$-factor variation indicates efficient, but
incomplete spin exchange averaging.
