Non-antisymmetric versions of Nambu-Poisson and algebroid brackets

We show that we can skip the skew-symmetry assumption in the definition of Nambu-Poisson brackets. In other words, an n-ary bracket on the algebra of smooth functions which satisfies the Leibniz rule and an n-ary version of t he Jacobi identity must be skew symmetric. A similar result holds for a non -antisymmetric version of Lie algebroids.