Magnetic skyrmions are localized non-collinear spin textures with a high potential for future spintronic applications. Skyrmion phases have been discovered in a number of materials and a focus of current research is to prepare, detect and manipulate individual skyrmions for implementation in devices. The local experimental characterization of skyrmions has been performed by, for example, Lorentz microscopy or atomic-scale tunnel magnetoresistance measurements using spin-polarized scanning tunnelling microscopy. 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.