Massive multiple-input multiple-output (MIMO) systems are a core technology designed to achieve the performance objectives defined for 5G wireless communications. They achieve high spectral efficiency, reliability, and diversity gain. However, the many radio frequency chains required in base stations equipped with a high number of transmit antennas imply high hardware costs and computational complexity. Therefore, in this paper, we investigate the use of a transmit-antenna selection scheme, with which the number of required radio frequency chains in the base station can be reduced. This paper proposes two efficient transmit-antenna selection (TAS) schemes designed to consider a trade-off between performance and computational complexity in massive MIMO systems. The spectral efficiency and computational complexity of the proposed schemes are analyzed and compared with existing TAS schemes, showing that the proposed algorithms increase the TAS performance and can be used in practical systems. Additionally, the obtained results enable a better understanding of how TAS affects massive MIMO systems.