Let $\sigma_i$ be a $n$-dimensional vector in $\mathbb{R}^n$. And $S$ a matrix where each row is the $i$-th $\sigma$ vector
\begin{equation} S = \begin{pmatrix} \sigma_1 \\ \sigma_2 \\ \vdots \\ \sigma_P \end{pmatrix} \end{equation}
Therefore I can write $R = S \times S^T$ as \begin{equation} R = S \times S^T = \begin{pmatrix} \langle \sigma_1 , \sigma_1 \rangle & % \langle \sigma_1 , \sigma_2 \rangle & % \cdots & \langle \sigma_1 , \sigma_P \rangle \\ \langle \sigma_2 , \sigma_1 \rangle & \langle \sigma_2 , \sigma_2 \rangle & \cdots & \langle \sigma_2 , \sigma_P \rangle \\ \vdots & \vdots & \ddots & \vdots \\ \langle \sigma_P , \sigma_1 \rangle & \langle \sigma_P , \sigma_2 \rangle & \cdots & \langle \sigma_P , \sigma_P \rangle \end{pmatrix} \end{equation}
To make things more visible, by dot product commutativity I can rewrite R as
\begin{equation} R = S \times S^T = \begin{pmatrix} \langle \sigma_1 , \sigma_1 \rangle & % \langle \sigma_1 , \sigma_2 \rangle & % \cdots & \langle \sigma_1 , \sigma_P \rangle \\ \langle \sigma_1 , \sigma_2 \rangle & \langle \sigma_2 , \sigma_2 \rangle & \cdots & \langle \sigma_2 , \sigma_P \rangle \\ \vdots & \vdots & \ddots & \vdots \\ \langle \sigma_1 , \sigma_P \rangle & \langle \sigma_2 , \sigma_P \rangle & \cdots & \langle \sigma_P , \sigma_P \rangle \end{pmatrix} \end{equation}
How can I describe $R^2$ without perform all calculations? Is there any way to generalize a rule to describe the $n$-th power of $R$?
By describe, I mean, write it down in matrix form without perform all calculations. Also If there is any properties that emerge that help to know the form and its elements.
For example, I know that a product of symmetric matrices are also symmetric, therefore I just need to calculate the upper triangular (or lower if I feel down). Any other "caveats"?