We give an explicit Stanley decomposition of the bracket ring \({\mathbb{B}}_{n,d}\), that is, the commutative ring generated by the \(d\times d\)-minors of a generic \(n\times d\)-matrix. A Stanley decomposition is a direct sum decomposition of the additive group of the ring, each summand of which is a bracket monomial times a subring generated freely by brackets. The decomposition is obtained via a shelling of the simplicial complex determined by the standard tableaux. Our construction has important applications in the Cushman-Sanders normal form theory for nilpotent vector fields.