Single magnetic atoms on surfaces are the smallest conceivable units for two-dimensional magnetic data storage. Previous experiments on such systems have investigated magnetization curves(1,2), the many-body Kondo effect(3,4) and magnetic excitations in quantum spin systems(5,6), but a stable magnetization has not yet been detected for an atom on a non-magnetic surface in the absence of a magnetic field. The spin direction of a single magnetic atom can be fixed by coupling it to an underlying magnetic substrate via the exchange interaction(7,8), but it is then difficult to differentiate between the magnetism of the atom and the surface. Here, we take advantage of the orbital symmetry of the spin-polarized density of states of single cobalt atoms to unambiguously determine their spin direction in real space using a combination of spin-resolved scanning tunnelling microscopy experiments and ab initio calculations. By laterally moving atoms on our non-collinear magnetic template(9), the spin direction can also be controlled while maintaining magnetic sensitivity, thereby providing an approach for constructing and characterizing artificial atomic-scale magnetic structures.