In this paper, we propose a geometry based algorithm for dynamical low-rank approximation on the manifold of fixed rank matrices. We first introduce a suitable geometric description of the set of fixed rank matrices which relies on a natural parametrization of matrices. More precisely, it is endowed with the structure of analytic principal bundle, with an explicit description of local charts. For matrix differential equations, we introduce a first order numerical integrator working in local coordinates. The resulting chart based algorithm can be interpreted as a particular splitting of the projection operator onto the tangent space of the low-rank matrix manifold. It is proven to be robust and exact in some particular case. Numerical experiments confirm these results and illustrate the behavior of the proposed algorithm.