In a course on supersymmetry I have learned the following way to construct spinor representations in $D$ dimensions: we first find matrices $\Gamma^A$ which obey the Clifford algebra $$\{\Gamma^A,\Gamma^B\}=2\delta^{AB}.$$
With these matrices in hand we define the new matrices $\Sigma^{AB}=\frac{1}{4}[\Gamma^A,\Gamma^B]$. We then verify that the $\Sigma^{AB}$ obey the Lie algebra of $\mathfrak{so}(D)$. Next to construct representations of the group one exponentiates the representation of the algebra, representing group elements as $\exp\left(\frac{1}{2}\theta_{AB}\Sigma^{AB}\right)$.
Now, while Physicists often do not make such a distinction, I'm aware that when we try to exponentiate the spinor representations, we cannot obtain a representation of ${\rm SO}(D)$. Instead we obtain a representation of its universal cover ${\rm Spin}(D)$.
Now why is that? Why this cannot give us a representation of ${\rm SO}(D)$ but rather it gives us a representation of ${\rm Spin}(D)$? What goes wrong for ${\rm SO}(D)$ which does not go wrong for ${\rm Spin}(D)$?