We propose a novel, near-optimal data detection algorithm and a corresponding FPGA design for large multiple-input multiple-output (MIMO) wireless systems. Our algorithm, referred to as TASER (short for triangular approximate semidefinite relaxation), relaxes the maximum-likelihood (ML) detection problem to a semidefinite program and solves a non-convex approximation using a preconditioned forward-backward splitting procedure. We show that TASER achieves near-ML performance at low computational complexity, even for large-dimensional MIMO systems. We develop a systolic array that implements TASER and achieves high throughput at low hardware complexity. To demonstrate the effectiveness of our solution, we develop reference designs on a Xilinx Virtex-7 FPGA for various antenna configurations. One of our TASER designs achieves up to 98Mb/s for a 32-user system that employs QPSK, while consuming only 150k FPGA look-up tables.