How to describe the variety for lines through the origin

Question

Let $I(x)$ be an ideal generated by the affine variety $X$. $I(X)=\langle xy,yz,zx\rangle$ gives the three coordinate axes through the origin (because it is the intersection of the $3$ planes $xy=0$, $yz=0$, and $zx=0$). If I want to have more than three lines through the origin, what would the generating ideal $I(X)$ and the variety be?