Show we can find an orthogonal transformation of determinant $1$ sending any point of $S^{n-1}$ into any other.
I searched online and could not find the solution. I am currently learning about Lie groups as part of a high school independent study project.
On
Not really an illuminating answer, but if you want $T(v) = w$ for an orthogonal transformation, then just extend $v$ to an orthonormal basis $A$ and similarly $w$ to an orthonormal basis $B$ and then your transformation matrix is $BA^{-1} = BA^T$. You need to show that inverses and products of orthogonal matrices are orthogonal to make this more rigorous, and argue that you can have $\det(AB^{-1}) = 1$ by swapping some vectors in $A$ or $B$ if necessary.
Among my course notes at http://www.math.umn.edu/~garrett/m/mfms/ meant to be as accessible as possible (with some more-sophisticated "supplements"), the case of rotations of spheres and some other examples are discussed in http://www.math.umn.edu/~garrett/m/mfms/06_homogeneous_geometries.pdf