If Ais a skew-symmetric matrix and n is odd positive integer, then Aⁿ is

Given Ais skew-symmetric. Hence,
$\begin{array}{l}{\mathrm{A}}^{\mathrm{T}}=-\mathrm{A}\\ \Rightarrow {\mathrm{A}}^{\mathrm{n}}={\left(-{\mathrm{A}}^{\mathrm{T}}\right)}^{\mathrm{n}}=-{\left({\mathrm{A}}^{\mathrm{T}}\right)}^{\mathrm{n}}=-{\left({\mathrm{A}}^{\mathrm{n}}\right)}^{\mathrm{T}}\text{ (given }\mathrm{n}\text{ is odd) }\end{array}$
Hence, Aⁿ is skew-symmetric