= Solution
Here the <orthogonal group> is the real <group> $O_n(\mathbb R)$. Differentiating $g(t)^Tg(t)=I$ at zero gives $X^T+X=0$. Conversely a real <skew-symmetric matrix> satisfies $(e^{tX})^Te^{tX}=e^{-tX}e^{tX}=I$. Therefore
$$
\boxed{L(O_n)=\{X\in M_n(\mathbb R):X^T=-X\},\qquad\dim_{\mathbb R}L(O_n)=n(n-1)/2}.
$$
For $n\ge2$ this is not a complex Lie subalgebra of $M_n(\mathbb C)$: multiplying a nonzero real member by $i$ leaves this real subspace. For $n=1$ the zero algebra is the trivial complex <Lie algebra> as well. The complex <orthogonal group> would be a different <group> from the real <orthogonal group> used here.
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