Let \(n\geq 1\). The special orthogonal group is \[ \SO_n(\RR) \coloneqq \theset{ A\in\Mat_{n\times n}(\RR) \st AA^t = I \text{ and } \det A = 1} = \Orth_n(\RR) \cap \SL_n(\RR). \]
Every \(A\in\Orth_n(\RR)\) satisfies \(\det A = \pm 1\), and \(\SO_n(\RR)\) is the kernel of \(\det\colon\Orth_n(\RR)\to\theset{\pm1}\), of index \(2\) in \(\Orth_n(\RR)\). For \(n\geq 2\), \(\SO_n(\RR)\) is a proper subgroup of \(\SL_n(\RR)\): the matrix \(\operatorname{diag}(2, 1/2, 1, \ldots, 1)\) lies in \(\SL_n(\RR)\) but not in \(\Orth_n(\RR)\).