As you may be aware, a skew-symmetric matrix, denoted as $A$, adheres to the equation $A^T=-A$. This equation provides a clear understanding of the structure of $A$, which can be represented as follows: $$A=\begin{bmatrix} 0 & a_{12} & \dots & a_{1n}\\ -a_{12} & 0 & \dots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ -a_{1n} & -a_{2n} & \dots & 0 \end{bmatrix}$$ I have come across a proof utilizing bilinear forms in the book by Hoffman and Kunze. Additionally, I discovered a proof employing determinants. These proofs are quite comprehensive and easy to understand. However, given that this problem was assigned as homework, I am restricted in the tools I can employ. I am in search of a more rudimentary proof that does not rely on determinants, polynomials, eigenvalues, etc. Based on the book by Hoffman and Kunze, I am permitted to use the following:
- Linear Equations (including key aspects such as elementary row operations, RREF form)
- Vector Spaces
- Linear Transformations (including key aspects such as representation of transformations by matrices, linear functionals, the double dual)
Any assistance or guidance would be greatly appreciated!
