Answer
See proof below.
Work Step by Step
Since $A$ is skew-symmetric, $A^{T}=-A$
We define the matrix $A$ as the following, $A=[a_{ij}]$. Thus
$-A=(-1)[a_{ij}]=[(-1)*a_{ij}]$
and $A^{T}=[a_{ji}]$
since $A^{T}=-A$, then
$[a_{ji}]=[(-1)*a_{ij}]$
$a_{ji}=(-1)*a_{ij}$
For the main diagonal elements $i=j$, therefore, we have
$a_{ii}=(-1)*a_{ii}$, thus $2* a_{ii}=0$ and therefore
$a_{ii}=0$
which indicates that the main diagonal elements of the matrix $A$ are zeros.
