Filippov $n$-algebroids are introduced by Grabowski and Marmo as a natural generalization of Lie algebroids. On this note, we characterized Filippov $n$-algebroid structures by considering certain multi-input connections, which we called Filippov connections, on the underlying vector bundle. Through this approach, we could express the $n$-ary bracket of any Filippov $n$-algebroid using a torsion-free type formula. Additionally, we transformed the generalized Jacobi identity of the Filippov $n$-algebroid into the Bianchi-Filippov identity. Furthermore, in the case of rank $n$ vector bundles, we provided a characterization of linear Nambu-Poisson structures using Filippov connections.