Show that the set of (2 n times 2 n) real matrices of the form [ left(begin{array}{rr}

Question:

Show that the set of $2 n \times 2 n$ real matrices of the form

\[
\left(\begin{array}{rr}
A & B \\
-B & A
\end{array}ight)
\]

is closed under matrix addition and multiplication. Show also that the 'linear' mapping into the space of complex $n \times n$ matrices

\[
\left(\begin{array}{rr}
A & B \\
-B & A
\end{array}ight) \mapsto A+i B
\]

is an isomorphism preserving matrix addition and multiplication.