Using spin-polarized scanning tunneling microscopy with ring electrodes, the in-plane spin polarization of Mn on Fe(001) was measured. A large (√{10} x 2√{10})R18.4^° reconstruction with a noncollinear spin structure was found. By combining maps of the spin polarization for two orthogonal in-plane directions, the vector field of the polarization in the unit cell could be constructed. The complex behavior is explained on the basis of the tendency of Mn to form antiferromagnetically coupled surface dimers.