In this paper, we propose a novel methodology to design optimal precoders for distributed detection of high-dimensional signals. We consider a wireless sensor network (WSN) that consists of multiple sensors that are spatially distributed in a region of interest and a fusion center (FC). The sensors observe an unknown high-dimensional signal and forward their observations to the FC after precoding. The sensors collect data over both temporal and spatial domains. The FC performs a binary hypothesis test based on the data received from the sensors over noisy channels. In this setup, we present a technique to design optimal online linear precoding strategies with transmit power constraints. We show analytically that the error exponents achieved by the proposed precoders are independent of the signal dimension. In contrast, the error exponents of the state-of-the-art precoding strategies deteriorate with the increase in signal dimension. We verify our analysis via numerical simulations and show that the proposed precoders achieve better detection performance compared to those of other state-of-the-art techniques known in the literature.