Ordered Basis for vector space groupoids

On vector spaces with an inner product, complex structure or symplectic form, it is possible to find ordered basis that adapted to those structures. It was not clear which basis one should choose for a vector space with a groupoid structure. In my work with Francesco Cattafi we choose some class of ordered basis, we proved this class is again a  groupoid, the matrices changing two such ordered basis has also a groupoid structure and generalize it for vector bundle groupoids.