# Bivector to pseudovector mapping

I am studying antisymmetric tensors and currently reading a topic on pseudovectors. I understand that every bivector can be mapped to a corresponding pseudovector and vice versa but it is mentioned that it can be done using the following mapping :

if we have $$\alpha \in \Lambda^{2} \mathbb{R^{3}}$$ which is a bivector with $$\alpha = \alpha^{23}e{_2}\land e{_3} + \alpha^{31}e{_3}\land e{_1} + \alpha^{12}e{_1}\land e{_2}$$ then we can define a one-one and onto mapping $$J$$ in component form as $$J : \Lambda^{2} \mathbb{R^{3}} \rightarrow \mathbb{R^{3}},$$

$$\alpha^{ij} \rightarrow (J(\alpha^{ij})) = \frac{1}{2} \epsilon^{i}{_j}{_k}\alpha^{jk}$$

Here I'm having trouble understanding what the form of the map $$J$$ is? Like I understand that the Levi Civita tensor of rank 3 has been used but can somebody explicitly show the action of $$J$$ (perhaps in matrix form) on this bivector and how it is transforming it to a vector?