Homotopy group for a certain subset of Euclidean space

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Let $X_{2,n} = \{ (x,y) \in \mathbb{R}^{n} \times \mathbb{R}^n : x \text{ and } y \text{ are linearly independent} \}$. How can I compute the homotopy group $\pi_2(X_{2,n})$ for $n \geq 3$?