We continue our study of orthogonal Lie algebras, i.e. Lie algebras which support an invariant scalar product. We first show that a Lie algebra has a non degenerate invariant bilinear form iff adjoint and coadjoint representations are isomorphic. We study the space of all invariant bilinear forms on an orthogonal algebra. In a second part we study orthogonal modules and give a complete description of the double extension process which allows to construct all orthogonal modules ; we give examples and raise the question of existence of non-isomorphic orthogonal structures on a given orthogonal Lie algebra.