Symmetric Matrix. $\mathcal{S}_{n}(\mathbb{R}) $ is the set of symmetric matrices of $ \mathcal{M}_{n}(\mathbb{R})$


*

*Show that if $ A \in \mathcal{S}_{n}(\mathbb{R})$, then: $ A = \sum_{i=1}^{n} a_{ii}E_{ii} + \sum_{1 \leq i < j \leq n}^{n}(2a_{ij})\big(\frac{1}{2}(E_{ji}+E_{ij}) \big)$


$E_{ii}$ is the elementary matrix.


*Show that $B_1 = (E_{ii})_{1 \leq i < n}  \cup (\frac{1}{2}(E_{ij} + E_{ji}))_{1 \leq i < j \leq n} $ is the basis of $\mathcal{S}_{n}(\mathbb{R}) $ and determine its demension 



$A$ is symmetric matrix: $ A = a_{ij} = a_{ji} = \sum_{i=1}^{n} a_{ij} E_{ij} = \sum_{i=1}^{n} a_{ji} E_{ji} $
We have: $ 2A = \sum_{i=1}^{n} a_{ij}(E_{ij} + E_{ji} ) $
I don't know how to proceed to get: $ A = \sum_{i=1}^{n} a_{ii}E_{ii} + \sum_{1 \leq i < j \leq n}^{n}(2a_{ij})\big(\frac{1}{2}(E_{ji}+E_{ij}) \big)$
I'm also stuck with the second question. 
Thank you. 

We can always decompose a matrix as a sum of elementary matrices, that is, 
$$A= \sum_{1\leqslant i,j\leqslant n}a_{i,j}E_{i,j}=\sum_{i=1}^na_{ii}E_{ii}+\sum_{1\leqslant i\neq j\leqslant n}a_{i,j}E_{i,j}.$$
The second sum can be written as 
$$\sum_{1\leqslant i\neq j\leqslant n}a_{i,j}E_{i,j}=\sum_{1\leqslant i\lt j\leqslant n}a_{i,j}E_{i,j}+\sum_{1\leqslant j\lt i\leqslant n}a_{i,j}E_{i,j}.$$ 
Now, use symmetry to write the second sum as $\sum_{1\leqslant j\lt i\leqslant n}a_{j,i}E_{i,j}$ and use a change of index. 
For the second question, use the first when to show that the family is generating. Then you have to show its linear independence. The case of dimension two can help.
