This work aims at providing a simple characterization of the geometric tangent cone to the Wasserstein space. The canonical construction in positively curved spaces builds from the set of geodesics up to an abstract closure in an appropriate topology. However, in the particular case of the Wasserstein space, it is further known that this abstract closure lies in the larger set of measures over the underlying tangent space. It is shown in this work that each member of this larger set is equivalent near~0 to its projection over the geometric tangent cone, which allows to understand the latter as a quotient structure. The argument relies on a seemingly new Helmholtz-Hodge decomposition for measure fields.