Matroids and semigraphoids are discrete structures abstracting and generalizing linear independence among vectors and conditional independence among random variables, respectively. Despite the different nature of conditional independence from linear independence, deep connections between these two areas are found and still undergoing active research. In this paper, we give a characterization of the embedding of matroids into conditional independence structures and its oriented counterpart, which lead to new axiom systems of matroids and oriented matroids.