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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?

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