Let \(\mathcal {A}\) be a von Neumann algebra acting on a complex Hilbert space \(\mathcal {H}\) and \(\mathcal {Z}_{S}(\mathcal {A})\) be the symmetric center of \(\mathcal {A}\) . It is shown that, if a map \(\phi : \mathcal {A}\rightarrow \mathcal {A}\) satisfies \([\phi (X), P]_{\diamond }=[X, \phi (P)]_{\diamond }\) for all \(X\in \mathcal {A}\) and all projections \(P\in \mathcal {A}\) , then there exists a map \(f: \mathcal {A}\rightarrow \mathcal {Z}_{S}(\mathcal {A})\) such that \(\phi (X)=X\phi (I)+f(X)\) for all \(X\in \mathcal {A}\) , where \(\phi (I)=\phi (I)^{*}\) and f satisfies \(f(P)=0\) for all projections \(P\in \mathcal {A}\) . Moreover, a characterization of nonadditive bi-skew commuting maps on von Neumann algebras is obtained
