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For a more interesting problem I wanted to see if I could find the basis of the vector space of all 3x3 skew symetric matricies.

Usually, I can find a general form for these types of problem. Such as the general form of a symetric matricie. But skew symetric matricies seem to have more than one form

[0 a b]

[-a 0 c]

[-b -c 0]

and

[0 a -b]

[-a 0 -c]

[b c 0]

I proved that this form of a skew symetric matrice is a basis

[0 a b]

[-a 0 c]

[-b -c 0]

but is it true for the vector space of all 3x3 skew symetric matricies of that form or all skew symetric matricies?