This paper addresses the design of transmit precoder and receive combiner matrices to support $N_{\rm s}$ independent data streams over a time-division duplex (TDD) point-to-point massive multiple-input multiple-output (MIMO) channel with either a fully digital or a hybrid structure. The optimal precoder and combiner design amounts to finding the top-$N_{\rm s}$ singular vectors of the channel matrix, but the explicit estimation of the entire high-dimensional channel would require significant pilot overhead. Alternatively, prior works suggest to find the precoding and combining matrices directly by exploiting channel reciprocity and by using the power iteration method, but its performance degrades in the low SNR regime. To tackle this challenging problem, this paper proposes a learning-based active sensing framework, where the transmitter and the receiver send pilots alternately using sensing beamformers that are actively designed as functions of previously received pilots. This is accomplished by using recurrent neural networks to summarize information from the historical observations into hidden state vectors, then using fully connected neural networks to learn the appropriate sensing beamformers in the next pilot stage and finally the transmit precoding and receive combiner matrices for data communications. Simulations demonstrate that the learning-based method outperforms existing approaches significantly and maintains superior performance even in the low SNR regime for both the fully digital and hybrid MIMO scenarios.