Let V be an arbitrary linear space and f : V × ... × V → V an n-linear map. We show that, for any choice of basis B of V, the n-linear map f induces on V a decomposition (depending on B) V = ⊕Vj as a direct sum of linear subspaces, which is f-orthogonal in the sense f(V,...,Vj,...,Vk,...,V) = 0 when j \neq k, and in such a way that any Vj is strongly f-invariant in the sense f(V,...,Vj,...,V) ⊂ Vj. We also characterize the f-simplicity of any Vj. Finally, an application to the structure theory of arbitrary n-ary algebras is also provided. It is the full generalization of some early result .