  A variational formulation is introduced for the Oseen equations written in
terms of vor\-ti\-city and Bernoulli pressure. The velocity is fully decoupled
using the momentum balance equation, and it is later recovered by a
post-process. A finite element method is also proposed, consisting in
equal-order N\'ed\'elec finite elements and piecewise continuous polynomials
for the vorticity and the Bernoulli pressure, respectively. The {\it a priori}
error analysis is carried out in the $\mathrm{L}^2$-norm for vorticity,
pressure, and velocity; under a smallness assumption either on the convecting
velocity, or on the mesh parameter. Furthermore, an {\it a posteriori} error
estimator is designed and its robustness and efficiency are studied using
weighted norms. Finally, a set of numerical examples in 2D and 3D is given,
where the error indicator serves to guide adaptive mesh refinement. These tests
illustrate the behaviour of the new formulation in typical flow conditions, and
they also confirm the theoretical findings.
