Skew-symmetric matrices

#linear_algebra

A real $n \times n$ matrix $A$ is called skew-symmetric if $A^{⊤} = - A$ . Let $V_{n}$ be the set of all skew-symmetric matrices in $M_{n} (R)$ . Recall that $M_{n} (R)$ is an $n^{2}$ -dimensional $R$ -vector space with standard basis ${e_{i j} | 1 \leq i, j \leq n}$ , where $e_{i j}$ is the $n \times n$ matrix with a 1 in the $(i, j)$ -position and zeros everywhere else.

Show $V_{n}$ is a subspace of $M_{n} (R)$ .
Find an ordered basis $B$ for the space $V_{3}$ of all skew-symmetric $3 \times 3$ matrices.

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A real $n\times n$ matrix $A$ is called {\bfseries skew-symmetric} if $A^{\top}=-A$. Let $V_n$ be the set of all skew-symmetric matrices in $\operatorname{M}_n({\bf R})$. Recall that $\operatorname{M}_n({\bf R})$ is an $n^2$-dimensional ${\bf R}$-vector space with standard basis $\left\{e_{ij}\,|\, 1\leq i,j\leq n\right\}$, where $e_{ij}$ is the $n\times n$ matrix with a 1 in the $(i,j)$-position and zeros everywhere else.
\begin{enumerate}[label=\alph*)]
	\item Show $V_n$ is a subspace of $\operatorname{M}_n({\bf R})$.
	\item Find an ordered basis $\mathcal{B}$ for the space $V_3$ of all skew-symmetric $3\times 3$ matrices.
\end{enumerate}