According to Wikipedia on Spin $1/2$ particles, the vectors of the $S_y$ basis, with respect to the $S_z$ basis, are: \begin{bmatrix} 0.707 \\ 0.707*i \end{bmatrix} \begin{bmatrix} 0.707 \\ -0.707*i \end{bmatrix}
I check their orthogonality by doing the dot product:
$0.707*0.707 + (0.707*i)*(-0.707*i) = 0.5 - 0.5*i^2 = 0.5 + 0.5 = 1$
Since it's not zero, the elements are not orthogonal. Doesn't that mean that they can't form a basis?
($S_z$ and $S_x$ are indeed orthogonal)