 Canonical orientation of a complex vector bundle

An ordered complex basis of a rank-$m$ complex vector space gives the real basis $(v_1,i{v}_1,\dots ,v_m,i{v}_m)$. Every complex change-of-basis matrix has positive real determinant $|{\det }_{\mathbb{C}}A{|}^2$, so these bases define a canonical integral orientation and hence an $R$-orientation for every commutative coefficient ring $R$.