Let $𝒳$ be a Banach space of dimension $n>1$ and $𝔄\subset ℬ\left(𝒳\right)$ be a standard operator algebra. In the present paper it is shown that if a mapping $d:𝔄\to 𝔄$ (not necessarily linear) satisfies $$d\left(\right[[U,V],W\left]\right)=\left[\right[d\left(U\right),V],W]+\left[\right[U,d\left(V\right)],W]+\left[\right[U,V],d(W\left)\right]$$ for all $U,V,W\in 𝔄$, then $d=\psi +\tau$, where $\psi$ is an additive derivation of $𝔄$ and $\tau :𝔄\to 𝔽I$ vanishes at second commutator $\left[\right[U,V],W]$ for all $U,V,W\in 𝔄$. Moreover, if $d$ is linear and satisfies the above relation, then there exists an operator $S\in 𝔄$ and a linear mapping $\tau$ from $𝔄$ into $𝔽I$ satisfying $\tau \left(\right[[U,V],W\left]\right)=0$ for all $U,V,W\in 𝔄$, such that $d\left(U\right)=SU-US+\tau \left(U\right)$ for all $U\in 𝔄$.