In each reference I check I am told, without proof, that orientable Riemannian manifolds have holonomy contained in SO$(n)$ by obviousness. I am sure I will feel bad about myself when I hear the answer given its alleged evidency, but this is not so clear to me (in terms of rigorous proof, the intuition is solid).
I have a similar confusion for Calabi-Yau manifolds. I see CYs defined (compact) Kählers having SU$(n)$ holonomy or as Kählers having trivial canonical bundle. I see that having trivial canonical bundle implies a non-vanishing global section, but I do not see why the existence of a non-vanishing holomorphic section of the canonical bundle pushes us from U$(n)$ to SU$(n).$
At any point $x$, the action of the holonomy group $\textrm{Hol}_x(\nabla)$ of the Levi-Civita connection $\nabla$ on $T_x M$ induces an action on $\Lambda^n T^*_x M$. Since the Levi-Civita connection preserves the metric and the orientation, it preserves the volume form $\textrm{vol}$ they define. So, the action of $\textrm{Hol}_x(\nabla)$ on $\Lambda^n T^*_x M$ preserves $\textrm{vol}_x$, and hence $\textrm{Hol}_x(\nabla)$ is contained in the stabilizer $\textrm{SL}(\textrm{vol}_x) \cong \textrm{SL}(n, \Bbb R)$ and thus in the intersection $\textrm{O}(g) \cap \textrm{SL}(\textrm{vol}_x) \cong \textrm{SO}(n, \Bbb R)$.