Generators of the orthogonal group

Question

The real orthogonal group $O(N) = \{ A \in GL(N,\mathbb{R})|A^TA = AA^T = 1 \}$ consists of rotations with $\det A = +1$ (forming the subgroup $SO(N)$) and of reflections with $\det A = -1$. Its algebra is given by the skew-symmetric matrices $\mathfrak{o}(N) = \{ G \in GL(N,\mathbb{R})|G^T = -G \}$. Therefore each element of $O(N)$ should be generated by an element of $\mathfrak{o}(N)$ via a matrix exponential \begin{equation} A = \exp G. \end{equation} It is easy to check that $A$ is indeed orthogonal. However, its determinant is always \begin{equation} \det A = \det \exp G = \exp \text{tr}\ G = +1, \end{equation} because skew-symmetric matrices are traceless $\text{tr}\ G = 0$. Therefore we only generate $SO(N)$. Where is my error?