Let's define "preserve orientation" in the following way (I am not sure it is right, pls point out if there is something wrong):
For a linear transformation, we only need to check non-parallel vectors, because parallel vectors naturally come out parallel under linear transformation.
A rotation is a transformation that preserves length, angle and orientation. Any transformation preserves length and angle must preserves the dot product, and by then it is a linear transformation, and the transformation matrix has to be orthogonal by definition & properties of orthogonal matrix.
Now my question is how to prove a rotation is exactly the orthogonal matrix of determinant 1, i.e. only such orthogonal matrix with determinant 1 preserves orientation?
I have a wild guess that a linear transformation $\bf T$ preserves orientation iff it has positive determinant, so we immediately have the above claim. But I don't know how to show it. This is a wild guess and very likely to be wrong. Thank you for you guys' help!