Help with tensor calculus identity proof (antisymmetric matrix and levi civita symbol)

Question

I'm having some trouble with 2 identitys from tensor calculus. I need to proof these two guys:

in euclidean 3-dimensional space, an antisymmetric matrix with entries $M_{ij}$ is equivalent to a vector $v^k=frac{1}{2}epsilon^{kij},,M_{ij}$
the inverse formula is $M_{ij}=frac{1}{2}epsilon_{ijk},,v^k$

I know that the levi civita symbol is totally  antisymmetric, and so any other totally  antisymmetric  object Mij will be proportional  to  the  levi-civita symbol, but I just cant see the two informations adding up.
I appreciate any hint!

joigus · Answer

The more ways you have to convince yourself that something is correct, the better. Try G. Smith's approach, then think that $M_{ij}$ and $v^k$ have the same number of independent components, as,
$$
frac{nleft(n-1right)}{2}=frac{3left(3-1right)}{2}=3
$$
And finally, an identity to consider that will help you solve this and, e.g., prove many identities of vector calculus is,
$$
epsilon_{ijk}epsilon^{kmn}=left.delta_{i}right.^{m}left.delta_{j}right.^{n}-left.delta_{i}right.^{n}left.delta_{j}right.^{m}
$$

Help with tensor calculus identity proof (antisymmetric matrix and levi civita symbol)

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