In this paper we investigate the application of Compressed Sensing (CS) to MIMO channel sounding in the spatial domain. A compressive spatial channel sounder is proposed and evaluated based on real scenarios showing advantages in terms of time, hardware complexity and resolution. In particular, in the case where we use time division duplex for measuring the MIMO channel (in the form of antenna switching at the transmitter and/or the receiver), the proposed approach reduces the total number of switching periods, which implies a reduced channel acquisition time and thus an improved Doppler bandwidth. Alternatively, if we use multiple receive RF chains for the measurement, the compression allows to reduce the number of RF chains, which is a relevant advantage in terms of the overall receiver complexity, the amount of data to be processed in the digital domain (e.g., FPGA), power consumption, as well as RF hardware calibration. On the other hand, for the same measurement time and/or hardware complexity, one can increase the number of array elements to cover a larger aperture and so achieving better performance in terms of resolution.