I'm following Gompf and Stipsicz book about $4$-Manifold and Kirby Calculus. Here (page 34) they claim that the spinor bundle $S\to X$ over a spin compact manifold $X$ of dimension $n$ (even) is a Clifford bundle. I'm trying to prove that.
I'm interested in showing that the Clifford action is skew-adjoint with respect to the hermitian inner product. As far as I understood the bundle of spinors is build in the following way. Let $\rho\colon \text{Spin}(n)\to M_{2^n}(\Bbb C)$ be the representation of the Clifford algebra as a matrix algebra (the authors claim it's an isomorphism), with a little abuse of notation we will use $\rho$ to denote the representation of $\text{Spin}(n)\subset \Bbb C L_n$, given by the restriction of the above isomorphism. Therefore the spinor bundle $S$ is $$ S = P_{\text{Spin}(n)} \times_{\rho} \Bbb C^{2^n} \to X$$ We have an action of $\Bbb C l (TX)$ on the second summand which respect the identifications, hence we have an action of the Clifford algebra on $S$. I need to prove that such an action is skew symmetric when restricted to unitary elements $v \in T_x(X)\subset \Bbb C l (TX)$. Skew-symmetry involves the fact that when restrict to a fiber of $S$ with the canonical hermitian product defined (please correct me if I'm wrong) as $$ \langle (\alpha,s_1),(\beta,s_2)\rangle := \langle \rho(\alpha)s_1,\rho(\beta)s_2\rangle$$ where the latter is the canonical hermitian product on $\Bbb C^{2^n}$, we have $$\langle v\cdot s_1,s_2\rangle = -\langle s_1,v\cdot s_2\rangle $$ for $s_1,s_2$ in some fiber of $S$.
Problem is that I don't understand what kind of property I can impose on the matrix $\rho(v)$ (which should be skew hermitian), and this prevents me for proving the result.
can someone help me with that?