We derive a class of bases, called straightening bases, for the vector space of polynomials over a doubly indexed set xij . of noncommuting variables. The elements of these bases are shuffles of minors of (noncommuting) determinants of the matrix (xij ). This result is analogous to the straightening bases for the commuting case derived by Doubilet, Rota, and Stein. We develop a general approach that includes both the commuting and the noncommuting case. Spanning properties are derived in a manner analogous to the commuting case. The key lemma is a combinatorial extension of a classical determinantal identity due to Schweins. The proof of linear independence is based on a systematic application of a bijection of Knuth, rather than through the Capelli operators used in the commuting case.