Intuition behind symmetric and antisymmetric tensors

For the start let's have a look at matrices.

Let $A \in \mathbb{R^{n\times n }}$ be any matrix.

Than $\text{sym}(A)=\frac{1}{2}(A+A^T)$ is it's symetric part and $\text{skew}(A)=\frac{1}{2}(A-A^T)$ its antisymetric part. Note that any matrix can be written as sum of its symetric and antisymetric part $A=\text{sym}(A) + \text{skew}(A)$

Now I explain why there is the factor $\frac{1}{2}$. If matrix $A$ is already symetric you want to have $\text{sym}(A) = A$. Without the factor you would get $\text{sym}(A) = 2A$. Similar for antisymetric part.

You can think about matrices as bilinear forms, $x^T A y = A(x,y)$. Matrix $A$ is symetric iff $A(x,y)=A(y,x)$ for all $x,y$. This brings as to idea to what symetric tensor might be. That $T$ multilinear form(or tensor) is symetric iff $T(...,x,...,y,...) = T(...,y,...,x,...)$ for all $x,y$ and for all positions where you preform the 'swap'. As with matrices you can take symetric part of tensor $T$:

$$\text{sym}(T)(x_1,...,x_n) = \frac{1}{n!} \sum_{\sigma\in S_n} T(x_{\sigma(1)},..,x_{\sigma(n)})$$

Again natural requirement is if you have symetric tensor $T$ than $\text{sym}( T)=T$, this explains the factor $\frac{1}{n!}$

And why bother with symetric and antisymetric tensors? Well I'll give few remarks from my studium.

In physics you encounter tensors quite a lot and they are often symetric or anytisymetric, like electromagnetic tensor is antisymmetric. Or in general relativity you often caclulate something like this $\sum_{ij} A_{ij}B_{ij}$. When $A$ is symmetric than $\sum_{ij} A_{ij}B_{ij}=\sum_{ij} A_{ij}\text{sym}(B)_{ij}$ which is useful.

Or in differential geometry you define differential forms which are antisymetric and thay are of great importance.

I hope I shed a little bit of light on this topic.