Is [math] \mathrm{SO}(2n,\mathbb{R}) \cap \mathrm{GL}(n,\mathbb{C}) [/math] connected?

I know [math] \mathrm{SO}(2n,\mathbb{R}),\mathrm{GL}(n,\mathbb{C}) [/math] individually are connected. But certainly in general the intersection of two connected Lie subgroup (of a connected Lie group) need not be connected.