What is the dimension of this Grassmannian? Why is  $2\times 3$ the dimension of $Gr_2(\mathbb{R}^5)$? and can one use the dimensions of Lie groups to derive this dimension?
Note: $Gr_2(\mathbb{R}^5)$ denotes the Grassmannian of all $2$-dimensional subspaces of $\mathbb{R}^5$.

I assume that by $Gr_2(\mathbb{R}^5)$ you mean the Grassmannian of 2-planes in $\mathbb{R}^5$. By introducing an inner product in $\mathbb{R}^5$ you can pass to orthogonal frames and deduce the isomorphism $Gr_2(\mathbb{R}^5)\simeq O(5)/(O(2)\times O(3))$, from which you get the answer.
