We present a detailed experimental and theoretical analysis of the spin dynamics of two-dimensional electron gases (2DEGs) in a series of n -doped GaAs Alx Ga1-x As quantum wells. Picosecond-resolution polarized pump-probe reflection techniques were applied in order to study in detail the temperature, concentration, and quantum-well-width dependencies of the spin relaxation rate of a small photoexcited electron population. A rapid enhancement of the spin lifetime with temperature up to a maximum near the Fermi temperature of the 2DEG was demonstrated experimentally. These observations are consistent with the D'yakonov-Perel' spin-relaxation mechanism controlled by electron-electron collisions. The experimental results and theoretical predictions for the spin relaxation times are in good quantitative agreement.