The tremendous bandwidth available in the millimeter wave (mmW) frequencies between 30 and 300 GHz have made these bands an attractive candidate for next-generation cellular systems. However, reliable communication at these frequencies depends extensively on beamforming with very high-dimensional antenna arrays. Estimating the channel sufficiently accurately to perform beamforming can thus be challenging both due to low coherence time and large number of antennas. Also, the measurements used for channel estimation may need to be made with analog beamforming where the receiver can "look" in only direction at a time. This work presents a novel method for estimation of the receive-side spatial covariance matrix of a channel from a sequence of power measurements made at different angular directions. The method reduces the spatial covariance estimation to a matrix completion optimization problem. To reduce the number of measurements, the optimization can incorporate the low-rank constraints in the channels that are typical in the mmW setting. The optimization is convex and fast, iterative methods are presented to solving the problem. Simulations are presented for both single and multi-path channels using channel models derived from real measurements in New York City at 28 GHz.