Construct self-adjoint operator taking x to y

Question

Given $x,y\in \mathbb{C}^n$, I need a self-adjoint operator $J$ such that $Jx=y$. It seems intuitive that such an operator would exist. We can always find some operator taking $x$ to $y$ (by letting $x$ be an element of some basis of $\mathbb{C}^n$ and then defining an operator on the basis elements, specifying that $x$ maps to $y$). But how can I construct this in a way that ensures the operator is self-adjoint?