This work analyzes the performance of a downlink multi-user multiple-input multiple-output (MU-MIMO) non-orthogonal multiple access (NOMA) communications system. To reduce hardware complexity and exploit antenna diversity, we consider a transmit antenna selection (TAS) scheme and equal-gain combining (EGC) receivers. Further, we consider Weibull-distributed fading channels to account for non-linearities of the propagation medium and to cover, as special cases, important fading scenarios such as Rayleigh and exponential models. Performance metrics such as the outage probability (OP) and the average bit error rate (ABER) are derived in an exact manner. An asymptotic analysis for the OP and for the ABER is also carried out. Moreover, we obtain exact expressions for the probability density function (PDF) and the cumulative distribution function (CDF) of the end-to-end signal-to-noise ratio (SNR). Interestingly, our results indicate that, except for the first user (nearest user), in a high-SNR regime the ABER achieves a performance floor that depends solely on the user's power allocation coefficient and on the type of modulation, and not on the channel statistics or the amount of transmit and receive antennas. To the best of the authors' knowledge, no performance analyses have been reported in the literature for the considered scenario. The validity of all our expressions is confirmed via Monte-Carlo simulations.