The inner product of Lie brackets $\langle \mu, \lambda\rangle = \sum\langle[e_i,e_j]_\mu,[e_i,e_j]_\lambda\rangle$ is independent of choice of basis

Question

According to page 6 of the paper “Ricci soliton solvmanifolds” by Lauret, the canonical inner product on $\mathbb R^n$ descends to an inner product on $V := \wedge^2 (\mathbb R^n)^* \otimes \mathbb R^n$ by the formula $$\langle\mu,\lambda\rangle = \sum\langle\mu(e_i,e_j),\lambda(e_i,e_j)\rangle=\sum\langle\mu(e_i,e_j),e_k\rangle\langle e_k,\lambda(e_i,e_j)\rangle.$$ How do I check that this definition is independent of choice of orthonormal basis?