A Willmore surface
$y:M\rightarrow S^{n+2}$
has a natural harmonic oriented conformal Gauss map
$Gr_{y}:M\rightarrow SO^{+}(1,n+3)/SO(1,3)\times SO(n)$
, which maps each point
$p\in M$
to its oriented mean curvature 2-sphere at
$p$
. An easy observation shows that all conformal Gauss maps of Willmore surfaces satisfy a restricted nilpotency condition, which will be called “strongly conformally harmonic.” The goal of this paper is to characterize those strongly conformally harmonic maps from a Riemann surface
$M$
to
$SO^{+}(1,n+3)/SO^{+}(1,3)\times SO(n)$
, which are the conformal Gauss maps of some Willmore surface in
$S^{n+2}.$
It turns out that generically, the condition of being strongly conformally harmonic suffices to be associated with a Willmore surface. The exceptional case will also be discussed.