Let $M$ be a 2-torsion free $\Gamma$-ring with left identity $e$. Let $D : M x M\rightarrow M$ be a symmetric bi-additive mapping and $d(x) = D(x, x)$. Let $\sigma$ and $\tau$ be an endomorphism and an epimorphism of $M$, respectively. We prove the following: \begin{itemize} em[(i)] if $d$ is $(\sigma ,\tau )$-skew-commuting on $M$, then $D = 0$; em[(ii)] if $d$ is $(\tau ,\tau )$-skew-centralizing on $M$, then $d$ is $(\tau ,\tau )$-commuting on $M$; em[(iii)] if $M$ is a 3-torsion free $\Gamma$-ring satisfying $x\alpha y\beta z=x \beta y \alpha z$ for all $x, y, z\in M$ and $\alpha , \beta \in \Gamma$, then 2-$(\sigma , \tau )$-commutingness of $d$ on $M$ implies its $(\sigma ,\tau )$-commutingness. \end{itemize}