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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?

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