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