We study the melting of skyrmions in a two-dimensional Heisenberg chiral magnet with biaxial Dzyaloshinskii-Moriya interactions. At zero temperature, these topological excitations may form a triangular crystal with long-range positional order. We show using large-scale Monte Carlo simulations that at small finite temperature, however, the skyrmions rather form a typical two-dimensional solid, in which the positional correlations decay with distance as power laws while the orientational correlations remain finite. At higher temperature, we observe a direct transition from this two-dimensional solid to a liquid with short-range correlations. This differs from generic two-dimensional homogeneous particle systems, where a hexatic phase is sandwiched between the solid and the liquid.