A How to calculate the matrix of a form?

This is screenshot from V.I Arnold's book on Classical mechanics. My question is how do we find matrix of any n-form. Detailed answer please.
Screenshot_2019-02-11-16-28-22.jpeg
 

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Orodruin

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There is no such thing as a "matrix of an n-form". The point is that a 2-form has components of the form ##\omega_{ij}## with two indices, that can be represented as the components of a matrix. This is not the case for a general ##n##-form, which has components with ##n## indices.
 
So,how to find matrix given in screenshot for 2 forms?
There is no such thing as a "matrix of an n-form". The point is that a 2-form has components of the form ##\omega_{ij}## with two indices, that can be represented as the components of a matrix. This is not the case for a general ##n##-form, which has components with ##n## indices.
 

Orodruin

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Its just a matrix containing the components ##\omega_{ij}## in the appropriate positions.
 
Its just a matrix containing the components ##\omega_{ij}## in the appropriate positions.
Can you explain it in brief? I am new to forms. I forgot where it was written in book.
 

Orodruin

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Are you familiar on how to write down the components of a rank 2 tensor in a matrix?
 

Orodruin

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So the wedge product for the dual basis is defined as
$$
dx^i \wedge dx^j = dx^i \otimes dx^j - dx^j\otimes dx^i.
$$
You can work from there.
 

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