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/AlGaAs 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 life-time with temperature up to a maximum near the Fermi temperature of the 2DEG was demonstrated experimentally. These observations are consistent with the Dyakonov-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.