In this article, we devise a variant of the extended Kalman filter that can be generally applied to systems on manifolds with simplicity and low computational cost. We extend a given system on a manifold to an ambient open set in Euclidean space and modify the system such that the extended system is transversely stable on the manifold. Then, we apply the standard extended Kalman filter derived in Euclidean space to the modified dynamics. This method is efficient in terms of computation and accurate in comparison with the standard extended Kalman filter. It has the merit that we can apply various Kalman filters derived in Euclidean space including extended Kalman filters for state estimation for systems defined on manifolds. The proposed method is successfully applied to the rigid body attitude dynamics whose configuration space is the special orthogonal group in three dimensions.