How to show $\hat {\partial_i u_j} = \frac{\xi_i \xi_k}{|\xi|^2} \hat {\omega_{kj}}$ for $i,j=1,2,3,...,n$

Question

Consider $u \in (L^2 (\mathbb R^n))^n$, i.e. $u$ is a vector-valued function. Suppose $\omega_{ij} = \partial_i u_j - \partial_j u_i$ for $i,j=1,2,3,...,n$. How to show $\hat {\partial_i u_j} = \frac{\xi_i \xi_k}{|\xi|^2} \hat {\omega_{kj}}$ for $i,j=1,2,3,...,n$? Here, the $\hat{}$ means the Fourier transform.