Magnetism in ultrathin (1-10 ML) Fe films grown on Cu(100) has been studied by spin-polarized secondary electron emission spectroscopy. The variation of the magnetization with temperature and oxygen adsorption was investigated for various film thicknesses. The orientation of the magnetization for films between 5 and 6 ML thick switches reversibly between perpendicular (at low temperature) to in-plane (at high temperature). The switching transition temperature decreases with increasing film thickness, and is accompanied by a loss of long-range order over a range of 20-30 K. The transition is attributed to the temperature dependence of the perpendicular anisotropy. The effect of oxygen adsorption onto films with perpendicular remanence is to first suddenly turn the magnetization into the plane at a critical coverage, and then to kill the magnetization gradually with continued exposure. This indicates that the uniaxial surface anisotropy at the Fe-vacuum interface plays a major role in the magnetization of the film.