The size and surface effects on the magnetism of a fcc Fe (001) surface was investigated by performing firstprinciples calculations on 3, 5, 7, and 9 monolayers fcc Fe (001) single slabs with two different two-dimensional lattice constants, ${alpha}=3.44{AA}$ (System I) and 3.65 ${AA}$ (System II), using the all-electron full-potential linearized augmented plane wave method within a generalized gradient approximation. The surface layers were coupled ferromagnetically to the subsurface layer in both systems. However, the magnetism of the inner layers was quite different from each other. While all the inner layers of System II were ferromagnetically coupled in the same way as the surface layer, the inner layers of System I showed a peculiar magnetism, bilayer antiferromagnetism. The calculated spin magnetic moments per Fe atom were approximately 2.7 and 2.9 ${mu}_B$ at the surface for Systems I and II, respectively, due to the almost occupied Fe d-state being in the majority spin state and band narrowing. The spin orientations of System I were out-of-plane regardless of its thickness, whereas the orientation of System II changed from out-of-plane to in-plane with increasing thickness.