I'm wondering again what I'm missing in my understanding. In Peskin and Schroeder, as well as in other sources, the spinor representation of Lorentz transformation is given by $$\Lambda_\frac{1}{2}=exp(-\frac{i}{2}\omega_{\mu\nu}S^{\mu\nu})$$. Where $\omega_{\mu\nu}$ is antisymmetric tensor representing the Lorentz transformation and $S^{\mu\nu}=\frac{i}{4}[\gamma^\mu,\gamma^\nu]$. In the same sources they write that $$\Lambda^\dagger_\frac{1}{2}=exp(\frac{i}{2}\omega_{\mu\nu}(S^{\mu\nu})^\dagger)$$. The question, using the matrix identity $(AB)^T=B^TA^T$ shouldn't we have $$\Lambda^\dagger_\frac{1}{2}=exp(\frac{i}{2}(S^{\mu\nu})^\dagger\omega_{\mu\nu})$$?