Chiral skyrmions are stable particle-like solutions of the Landau-Lifshitz equation for ferromagnets with the Dzyaloshinskii-Moriya interaction (DMI), characterized by a topological number. We study the profile of an axially symmetric skyrmion and give exact formulas for the solution of the corresponding far field and near field equations, in the asymptotic limit of small DMI constant (alternatively large anisotropy). The matching of these two fields leads to a formula for the skyrmion radius as a function of the DMI constant. The derived solutions show the different length scales which are present in the skyrmion profiles. The picture is thus created of a chiral skyrmion that is born out of a Belavin-Polyakov solution with an infinitesimally small radius, as the DMI constant is increased from zero. The skyrmion retains the Belavin-Polyakov profile over and well-beyond the core before it assumes an exponential decay; the profile of an axially-symmetric Belavin-Polyakov solution of unit degree plays the role of the universal core profile of chiral skyrmions.