Let R be a ring with left identity. Let G : $R{ imes}R{ o}R$ be a symmetric biadditive mapping and g the trace of G. Let ${alpha};:;R{ o}R$ be an endomorphism and ${eta};:;R{ o}R$ an epimorphism. In this paper we show the following: (i) Let R be 2-torsion-free. If g is (${alpha},{eta}$)-skew-commuting on R, then we have G = 0. (ii) If g is (${eta},{eta}$)-skew-centralizing on R, then g is (${eta},{eta}$)-commuting on R. (iii) Let $n{ge}2$. Let R be (n+1)!-torsion-free. If g is n-(${alpha},{eta}$)-skew-commuting on R, then we have G = 0. (iv) Let R be 6-torsion-free. If g is 2-(${alpha},{eta}$)-commuting on R, then g is (${alpha},{eta}$)-commuting on R.