Say I have a rotation matrix $C(q) \in R^{3\times3}$ which is a function of a Hamilton quaternion $q=[q_w, q_x, q_y, q_z]^T=[q_w, q_v]^T$ where $q_v$ is the vector part of the quaternion. The Direction cosine matrix is defined as $$R=(q_w^2-q_v^Tq_v)I+2q_v^Tq_v+2q_w[q_v]_\times$$
where $[q_v]_\times$ is the skew symmetric matrix.
Is it correct to write: $$\left[C(q)-C(\hat{q})\right]p=C(q-\hat{q})p$$
where $p \in R^3$
Is this correct in the case of a small difference between $q$ and $\hat{q}$?