The transverse function approach, a control design method initially developed by the authors for nonlinear driftless systems, is based on a theorem that establishes the equivalence between the satisfaction of the Lie algebra rank condition (LARC) by a family of vector fields and the existence of functions “transverse” to these vector fields, defined on a torus of adequate dimension. In this paper we prove the existence of transverse functions defined on special orthogonal groups for vector fields which satisfy the LARC at the order one, and we provide an explicit expression of such functions. An example illustrates some of the advantages resulting from the use of these new functions.