Magnetic skyrmions are localized non-collinear spin textures with a high potential for future spintronic applications(1-12). Skyrmion phases have been discovered in a number of materials(9,11) and a focus of current research is to prepare, detect and manipulate individual skyrmions for implementation in devices(6-8). The local experimental characterization of skyrmions has been performed by, for example, Lorentz microscopy(3) or atomic-scale tunnel magnetoresistance measurements using spin-polarized scanning tunnelling microscopy(4,7,12). Here we report a drastic change of the differential tunnel conductance for magnetic skyrmions that arises from their non-collinearity: mixing between the spin channels locally alters the electronic structure, which makes a skyrmion electronically distinct from its ferromagnetic environment. We propose this tunnelling non-collinear magnetoresistance as a reliable all-electrical detection scheme for skyrmions with an easy implementation into device architectures.