Let $y\in \Bbb{R^n},$ where $n\geq 2$ so that $||y||=1.$ Here $||\cdot||$ is the Euclidean norm. Define the subspace $S=\{x\in \Bbb{R^n},x^{T}y=0\}.$ For $1\leq i\neq j\leq n$, define $v^{(i,j)}\in \Bbb{R^n}$ as follows:
$$v_{k}^{(i,j)}=\begin{cases}-y_{j},& \;k=i;\\\;\;y_{i},& \;k=j;\\\;\;0,& \;otherwise\end{cases}$$
I want to show that $S=\operatorname{span}\{v^{(i,j)},\;1\leq i\neq j\leq n\}.$ Note that $x_j$ denote the $jth$ component of a vector $x.$ I'm shut of ideas! Can anyone help out?