The spin-polarized scanning tunneling microscope (SP-STM) operated in the constant current mode is proposed as a powerful tool to investigate complex atomic-scale magnetic structures of otherwise chemically equivalent atoms. The potential of this approach is demonstrated by successfully resolving the magnetic structure of Cr/Ag(111), which is predicted on the basis of ab initio vector spin-density calculations to be a coplanar noncollinear periodic 120 degrees Néel structure. Different operating modes of the SP-STM are discussed on the basis of the model of Tersoff and Hamann.