Coherent electron spin dynamics in 10-nm-wide InGaAs/InAlAs quantum wells is studied from 10 K to room temperature using time-resolved Kerr rotation. The spin lifetime exceeds 1 ns at 10 K and decreases with temperature. By varying the spatial overlap between pump and probe pulses, a diffusive velocity is imprinted on the measured electron spins and a spin precession in the spin-orbit field is measured. A Rashba symmetry of the spin-orbit interaction is determined. By comparing the spatial precession frequency gradient with the spin decay rate, an upper limit for the Rashba coefficients α of 2 × 10-12 eVm is estimated.