  Let ${\mathcal M}$ be a von Neumann algebra without central summands of type
$I_1$. Assume that $\Phi:{\mathcal M}\rightarrow {\mathcal M}$ is a surjective
map. It is shown that $\Phi$ is strong skew commutativity preserving (that is,
satisfies $\Phi(A)\Phi(B)-\Phi(B)\Phi(A)^*=AB-BA^*$ for all $A,B\in{\mathcal
M}$) if and only if there exists some self-adjoint element $Z$ in the center of
${\mathcal M}$ with $Z^2=I$ such that $\Phi(A)=ZA$ for all $A\in{\mathcal M}$.
The strong skew commutativity preserving maps on prime involution rings and
prime involution algebras are also characterized.
