  We perform a full microscopic investigation on the spin relaxation in
$n$-type (001) GaAs quantum wells with Al$_{0.4}$Ga$_{0.6}$As barrier due to
the D'yakonov-perel' mechanism from nearly 20 K to the room temperature by
constructing and numerically solving the kinetic spin Bloch equations. We
consider all the relevant scattering such as the electron--acoustic-phonon, the
electron--longitudinal-optical-phonon, the electron--nonmagnetic-impurity and
the electron-electron Coulomb scattering to the spin relaxation. The spin
relaxation times calculated from our theory with a fitting spin splitting
parameter are in good agreement with the experimental data by Ohno {\em et al.}
[Physica E {\bf 6}, 817 (2000)] over the whole temperature regime (from 20 K to
300 K). The value of the fitted spin splitting parameter agrees with many
experiments and theoretical calculations. We further show the temperature
dependence of the spin relaxation time under various conditions such as
electron density, impurity density and well width. We predict a peak solely due
to the Coulomb scattering in the spin relaxation time at low temperature ($<50$
K) in samples with low electron density ({\em e.g.}, density less than $1
\times 10^{11}$ cm$^{-2}$) but high mobility. This peak disappears in samples
with high electron density ({\em e.g.} $2 \times 10^{11}$ cm$^{-2}$) and/or low
mobility. The hot-electron spin kinetics at low temperature is also addressed
with many features quite different from the high temperature case predicted.
