In special relativity, the underlying manifold is the vector space $\mathbb R^4$ equipped with the Minkowski metric. The position of a (point) particle located at $p\in\mathbb R^4$ is the point $p$ itself, at least with respect to the standard basis. A reference frame moving with constant speed $v$ is then another basis related to the standard basis by the Lorentz transformation (or the inverse thereof).
In General relativity we have a general manifold equipped with some Lorentzian metric. For a particle at a point $p$ on the manifold, do we define the position vector locally using some tetrad at the tangent space at $p$? And if the particle is moving, do we keep on choosing a tetrad at each point in order to define the position vector or can we define a tetrad field which smoothly varies with the particle so that the position vector is a smooth vector field? If so, is it possible to define the tetrad globally? Or is there another way of dealing with the situation of defining position vectors in GR?
Thank you.