For any pair of permutations, Macdonald defines a skew divided difference operator and shows how these operators can be used to compute the structure constants for Schubert polynomials. We will show that any skew divided difference operator can be written explicitly as a polynomial in the degree 1 divided difference operators with positive coefficients, which settles a conjecture of Kirillov. The proof relies on various tools from the braided Hopf algebra structure of the Fomin-Kirillov algebra.