What would be an antisymmetric bilinear form on $\mathbb{R^{4}}$ that cannot be written as a wedge product of two covectors? Also, what would be a $4\times4$ skew-symmetric matrix of rank $4$? Finally, how would I show that every antisymmetric bilinear form on $\mathbb{R^{4}}$ can be written as a finite linear combination of wedge products of covectors?
Any help would be appreciated, I'm really not sure how to start. Thank you!
