  Magnetic skyrmions are two-dimensional non-collinear spin textures
characterised by an integer topological number. They commonly crystallise at
low temperatures in bulk noncentrosymmetric ferromagnets where the lack of
inversion symmetry leads to an antisymmetric component of the exchange
interaction. Recently, stable room-temperature skyrmions were reported in
stacks of thin magnetic films where antisymmetric exchange results from broken
symmetry at the interface. Determining the spin structure of these
technologically-relevant skyrmions in the presence of external magnetic fields
has remained a key experimental challenge. Here, we use the single electron
spin of a Nitrogen-Vacancy (NV) centre in diamond to perform scanning
magnetometry of skyrmions in Pt/Co/Ta multilayers under ambient conditions. We
introduce a novel method to assess the manifold of spin textures compatible
with the measured local magnetic fields and identify physically allowed
configurations based on the topology of the resulting solution. We determine
that the underlying magnetization pattern for the skyrmion is consistent with a
cycloid (or N\'eel)-like spin texture, in agreement with theoretical
predictions for interfacial antisymmetric exchange interaction. Our results
open up wide-ranging possibilities in quantitative model-free imaging of
two-dimensional spin structures using scanning NV magnetometry.
