Prove: Square Matrix Can Be Written As A Sum Of A Symmetric And Skew-Symmetric Matrices Let $C^{n \times n}$ be a square matrix.

Prove that $$C=\frac{1}{2}(C+C^T)+\frac{1}{2}(C-C^T)$$

What I have manage so far is:

a. Let $S$ be a Symmetric Matrix so $S=C+C^T$ 

b. Let $N$ be a Skew-Symmetric Matrix so $N=C-C^T$ 

Proof:

$S^t=[C+C^T]^T=C^T+C=S$

$N^t=[C-C^T]^T=-C^T+C=-N$
If $A$ is a square matrix in $\mathcal M_n(\Bbb C)$ then
$$A=\underbrace{\frac12(A+A^t)}_{\text{symmetric matrix}}+\underbrace{\frac12(A-A^t)}_{\text{skew-symmetric matrix}}$$
Using indicial notation, show that the square matrix can be expressed as the sum of a symmetric and antisymmetric matrix by the decomposition:


  
    $$A=\frac{A+A^T}{2} + \frac{A-A^T}{2}$$