# Killing vectors corresponding to the Lorentz transformations

I have a problem with one thing.

Let's consider the Lorentz group and the vicinity of the unit matrix. For each $$\hat{L}$$ from such vicinity one can prove that there exists only one matrix $$\hat{\epsilon}$$ such that $$\hat{L}=exp[\hat{\epsilon}]$$. If we take $$\epsilon^{\mu \nu},\ \mu < \nu$$, as the parameters on the Lorentz group in a vicinity of the unit matrix, then we can compute the corresponding Killing vectors as $$\xi_{\alpha \beta}^{\mu}=\frac{\partial x^{'\mu}}{\partial \epsilon^{\alpha \beta}}\left(\hat{\epsilon}=0 \right)$$ where $$x^{'\mu}=L^{\mu}_{\nu}x^{\nu}$$. Here is my problem: during the computations there is one line that I do not get, namely: $$\cfrac{\partial L^{\mu \nu}}{\partial \epsilon^{\alpha \beta}}\left(\hat{\epsilon}=0 \right)=\delta^{\mu}_{\alpha}\delta^{\nu}_{\beta}-\delta^{\mu}_{\beta}\delta^{\nu}_{\alpha}$$. The $$(\epsilon^{\mu \nu})$$ matrix is antisymmetric, so that is why we get the difference of the Kronecer delta?