# Matrix representation of complex structures

Let $$J$$ be a matrix representation of a linear complex structure ($$J^2= -I$$, where $$I$$ is the identity matrix). In the case of the $$2\times 2$$ real matrices a general complex structure can be written as: $$J=\begin{bmatrix} a & b\sqrt{1+a^2} \\ -\frac{1}{b}\sqrt{1+a^2} & -a \\ \end{bmatrix}.$$ Is there a similar parametrization for higher dimensional matrices? In particular for a $$4\times 4$$ matrix?