Geometrical meaning of $ \begin{pmatrix} a\\ b \end{pmatrix} \mapsto \begin{pmatrix} a&-b\\ b&a \end{pmatrix}$ 
Geometrical meaning of $\tau: \mathbb{R^2} \rightarrow \mathbb{R^{2,2}}, \begin{pmatrix} a\\ b \end{pmatrix} \mapsto \begin{pmatrix} a&-b\\ b&a \end{pmatrix}$

I have troubles of reading matrices and interpreting them. How can you interpret them? Here my best guess is, that a vector is mapped into a higherdimensional space. In the image, do I have to look at the columns to see the vectors, does that even make sense or any of the images below? 
EDIT:


The space $\mathbb{R}^{2,2}$ is isomorphic (as a vector space over $\mathbb{R}$) to $\mathbb{R}^4$, and there are no simple way to graphically represent this four dimensional space.
You can interpret geometrically the matrix as the linear transformation from $\mathbb{R}^2$ to $\mathbb{R}^2$, that transforms the basis vector $[1,0]^T$ to the vector $[a,b]^T$ and $[0,1]^T$ to $[-b,a]$, and note that these two vectors are orthogonal.
