A reconfigurable intelligent surface (RIS) can be used to improve the channel gain between a base station (BS) and user equipment (UE), but only if its $N$ reflecting elements are configured properly. This requires accurate estimation of the cascaded channel from the UE to the BS through each RIS element. If the channel structure is not exploited, pilot sequences of length $N$ must be used, which is a major practical challenge since $N$ is typically at the order of hundreds. To address this problem without requiring user-specific channel statistics, we propose a novel estimator, called reduced-subspace least squares (RS-LS) estimator, that only uses knowledge of the array geometry. The RIS phase-shift pattern is optimized to minimize the mean-square error of the channel estimates. The RS-LS estimator largely outperforms the conventional least-squares estimator, and can be utilized with a much shorter pilot length since it exploits the fact that the array geometry confines the possible channel realizations to a reduced-rank subspace.