  The robustness of the gapless topological surface state hosted by a 3D
topological insulator against perturbations of magnetic origin has been the
focus of recent investigations. We present a comprehensive study of the
magnetic properties of Fe impurities on a prototypical 3D topological insulator
Bi$_2$Se$_3$ using local low temperature scanning tunneling microscopy and
integral x-ray magnetic circular dichroism techniques. Single Fe adatoms on the
Bi$_2$Se$_3$ surface, in the coverage range $\approx 1%$ are heavily relaxed
into the surface and exhibit a magnetic easy axis within the surface-plane,
contrary to what was assumed in recent investigations on the opening of a gap.
Using \textit{ab initio} approaches, we demonstrate that an in-plane easy axis
arises from the combination of the crystal field and dynamic hybridization
effects.
