Answer
See proof below.
Work Step by Step
Since $A$ and $B$ are skew-symmetric matrices, then $A^{T}=-A$ and $B^{T}=-B$
Thus
$(A+B)^{T}=A^{T}+B^{T}=-A+(-B)=(-1)A+(-1)*B=(-1) * (A+B)=-(A+B)$
and $A+B$ is a skew-symmetric matrix.
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