Answer:
We have, A is a skew-symmetric matrix. \[\therefore \] \[{{A}^{T}}=-A\] \[\Rightarrow \] \[|{{A}^{T}}|\,=\,|-A|\] \[\Rightarrow \] \[|{{A}^{T}}|\,={{(-\,1)}^{n}}|A|\] \[[\because \,\,\,|KA|\,={{K}^{n}}|A|]\] \[\Rightarrow \] \[|A|\,={{(-\,1)}^{n}}|A|\] \[[\because \,\,\,|{{A}^{T}}|\,=\,|A|]\] \[\Rightarrow \] \[|A|\,=-|A|\] \[[\because \,\,\,n\,\,\text{is}\,\,\text{odd}]\] \[\Rightarrow \] \[2|A|=0\] \[\Rightarrow \] \[|A|\,\,=0\] Hence, the determinant of a skew-symmetric matrix of odd order is zero.
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