$A(t)$ is a continuous mapping from $\mathbb R$ to $GL(\mathbb R^{2n})$ such that $A^2(t)=-id$ for all $t$. Is there a $\epsilon>0$ and a continuous mapping $B(t)$ from $(-\epsilon,\epsilon)$ to $GL(\mathbb R^{2n})$, such that
${B^{ - 1}}(t)A(t)B(t) = \left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 0&{ - 1}\\ 1&0 \end{array}}& \cdots &0&0\\ \vdots & \ddots & \vdots & \vdots \\ 0& \cdots &0&{ - 1}\\ 0& \cdots &1&0 \end{array}} \right)$ for all $t\in (-\epsilon,\epsilon)$