Given $$ {\bf q} = \left[ {\matrix{ {\sin \theta \sin \phi } \cr {\sin \theta \cos \phi } \cr {\cos \theta } \cr } } \right] $$ then $$ d{\bf q} = \left[ {\matrix{ {\cos \theta \sin \phi } \cr {\cos \theta \cos \phi } \cr { - \sin \theta } \cr } } \right]d\theta + \left[ {\matrix{ {\sin \theta \cos \phi } \cr { - \sin \theta \sin \phi } \cr 0 \cr } } \right]d\phi $$ and of course, being $\bf q$ unitary, we will have: $$ {\bf q} \cdot d{\bf q} = 0 $$
So $|d \bf q|$ represents the rotation angle, which is on the plane containing $\bf q$ and $\bf q+d\bf q$ and any linear combination of them, thus also $\bf q$ and $d \bf q$, which are normal to each other. Therefore their cross product is normal to the rotation plane and is equal to $$ \eqalign{ & {\bf w} = {\bf q} \times d{\bf q} = \left( {\left| {\bf q} \right|\left| {d{\bf q}} \right|\sin \left( {\mathop {{\bf q},d{\bf q}}\limits^ \wedge } \right)} \right)\;{\bf n} = \left| {d{\bf q}} \right|\;{\bf n} = {d\alpha} \; {\bf n} \cr & = \left[ {\matrix{ {\sin \theta \sin \phi } \cr {\sin \theta \cos \phi } \cr {\cos \theta } \cr } } \right] \times \left[ {\matrix{ {\cos \theta \sin \phi } \cr {\cos \theta \cos \phi } \cr { - \sin \theta } \cr } } \right]d\theta + \left[ {\matrix{ {\sin \theta \sin \phi } \cr {\sin \theta \cos \phi } \cr {\cos \theta } \cr } } \right]\left[ {\matrix{ {\sin \theta \cos \phi } \cr { - \sin \theta \sin \phi } \cr 0 \cr } } \right]d\phi = \cr & = \left[ {\matrix{ { - \cos \phi } \cr {\sin \phi } \cr 0 \cr } } \right]d\theta + {1 \over 2}\left[ {\matrix{ {\sin \phi \sin (2\theta )} \cr {\cos \phi \sin (2\theta )} \cr {\cos (2\theta ) - 1} \cr } } \right]d\phi \cr} $$