Massive multiple-input multiple-output (MIMO) is a promising approach for cellular communication due to its energy efficiency and high achievable data rate. These advantages, however, can be realized only when channel state information (CSI) is available at the transmitter. Since there are many antennas, CSI is too large to feed back without compression. As a sparsifying basis, prior work has applied compressive sensing (CS) techniques with the two-dimensional discrete cosine transform (2D-DCT) and the instantaneous Karhunen-Loeve transform (KLT). 2D-DCT fails, however, to reflect the spatial correlation and channel conditions. Instantaneous KLT requires perfect CSI, which means it is not feasible in practice. In this paper, we propose a new sparsifying basis that reflects the long-term characteristics of the channel and a new reconstruction algorithm for CS. We also suggest that dimensionality reduction is more proper to compress, and compare performance with the conventional method. Numerical results confirm that the proposed channel feedback mechanisms show better performance in point-to-point (single user) and point-to-multi-point (multiuser) scenarios.