Linear span of $SO(n)$ Problem:

Describe the linear span of $SO(n, \mathbb{R})$ over $\mathbb{R}$ with a system of equations

I managed to solve it for $n=2$, where the matrices $A = [a_{ij}]$ have to satisfy $a_{11} = a_{22}, a_{12} =- a_{21}$.
For $n=3$ I presume $a_{ij} = -a_{ji}$ for $i \neq j$, but can't show that any matrix in $SO(3)$ these equality holds.
Can you help me please?
 A: The linear span of $SO(n,\mathbb R)$? When $n\ne 2$, that is simply all of $\mathbb R^{n\times n}$:
For $n=3$ we have, for example,
$$ \begin{bmatrix}0&0&0\\1&0&0\\0&0&0\end{bmatrix} = \frac12 \begin{bmatrix}&-1\\1\\&&1\end{bmatrix} + \frac12 \begin{bmatrix}&1\\1\\&&-1\end{bmatrix} $$
and there are similar constructions for all of the other diagonal and off-diagonal entries.
For $n=4$ you can use the same construction, except that it may not cancel out all unwanted diagonal elements. But this can be corrected for, using multiples and permutations of
$$ \begin{bmatrix}4\\&0\\&&0\\&&&0\end{bmatrix} = I +
\begin{bmatrix}1\\&1\\&&-1\\&&&-1\end{bmatrix} +
\begin{bmatrix}1\\&-1\\&&1\\&&&-1\end{bmatrix} +
\begin{bmatrix}1\\&-1\\&&-1\\&&&1\end{bmatrix}
$$
And this generalizes nicely to $n>4$. (As well as to any base field of characteristic $\ne 2$).
A: I dont think the last thing you wrote is correct. Maybe try the following "a matrix in $SO(n)$ has the property that, if you multiply it by its transpose you get the identity". The transpose sends $a_{ij}$ to $a_{ji}$. you should also be able to say what matrix multiplication is in terms of the $a_{ij}$. Multiplying the matrix and its transpose together and setting equal to the identity, you should get the equations you want
