In the circuit model, quantum computers rely on the availability of a universal quantum gate set. A particularly intriguing example is a set of two-qubit only gates: matchgates, along with SWAP (the exchange of two qubits). In this paper, we show a simple decomposition of arbitrary matchgates into better known elementary gates, and implement a matchgate in a linear-optics experiment using single photons. The gate performance was fully characterized via quantum process tomography. Moreover, we represent the resulting reconstructed quantum process in a novel way, as a fidelity map in the space of all possible nonlocal two-qubit unitaries. We propose the non-local distance - which is independent of local imperfections like uncorrelated noise or uncompensated local rotations - as a new diagnostic process measure for the non-local properties of the implemented gate.