‎Corresponding to the concept of $p$-angular distance $\alpha_p[x,y]:=\left\lVert\lVert x\rVert^{p-1}x-\lVert y\rVert^{p-1}y\right\rVert$‎, ‎we first introduce the notion of skew $p$-angular distance $\beta_p[x,y]:=\left\lVert \lVert y\rVert^{p-1}x-\lVert x\rVert^{p-1}y\right\rVert$ for non-zero elements of $x‎, ‎y$ in a real normed linear space and study some of significant geometric properties of the $p$-angular and the skew $p$-angular distances‎. ‎We then give some results comparing two different $p$-angular distances with each other‎. ‎Finally‎, ‎we present some characterizations of inner product spaces related to the $p$-angular and the skew $p$-angular distances‎. ‎In particular‎, ‎we show that if $p>1$ is a real number‎, ‎then a real normed space $\mathcal{X}$ is an inner product space‎, ‎if and only if for any $x,y\in \mathcal{X}\smallsetminus{\lbrace 0\rbrace}$‎, ‎it holds that $\alpha_p[x,y]\geq\beta_p[x,y]$‎.

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