  We have used Brillouin Light Scattering spectroscopy to independently
determine the in-plane Magneto-Crystalline Anisotropy and the
Dzyaloshinskii-Moriya Interaction (DMI) in out-of-plane magnetized
Au/Co/W(110). We found that the DMI strength is 2-3 times larger along the
bcc$[\bar{1}10]$ than along the bcc$[001]$ direction. We use analytical
considerations to illustrate the relationship between the crystal symmetry of
the stack and the anisotropy of microscopic DMI. Such an anisotropic DMI is the
first step to realize isolated elliptical skyrmions or anti-skyrmions in thin
film systems with $C_{2v}$ symmetry.
