  We have developed a sensitive cryogenic second-harmonic generation microscopy
to study a van der Waals antiferromagnet MnPS$_3$. We find that long-range
N\'eel antiferromagnetic order develops from the bulk crystal down to the
bilayer, while it is absent in the monolayer. Before entering the long-range
antiferromagnetic ordered phase in all samples, an upturn of the second
harmonic generation below 200 K indicates the formation of the short-range
order and magneto-elastic coupling. We also directly image the two antiphase
(180$^{\circ}$) antiferromagnetic domains and thermally-induced domain
switching down to bilayer. An anomalous mirror symmetry breaking shows up in
samples thinner than ten layers for the temperature both above and below the
N\'eel temperature, which indicates a structural change in few-layer samples.
Minimal change of the second harmonic generation polar patterns in strain
tuning experiments indicate that the symmetry crossover at ten layers is most
likely an intrinsic property of MnPS$_3$ instead of an extrinsic origin of
substrate-induced strain. Our results show that second harmonic generation
microscopy is a direct tool for studying antiferromagnetic domains in
atomically thin materials, and opens a new way to study two-dimensional
antiferromagnets.
