In three dimensions two planes are orthogonal when their normal vectors are orthogonal (their inner product is zero). For example, planes $xy$ and $xz$ are orthogonal because the vectors $\hat{z}$ and $\hat{y}$ which are normal to the planes, respectively, are orthogonal, i.e $\hat{z}\cdot \hat{y}=0$.
How we define orthogonality of planes in $n$ dimensions? I am talking about 2d planes through the origin, in n-dimensional Euclidean space, that are specified by orthonormal vectors $\hat{x}_1, \hat{x}_2,.., \hat{x}_n$.
In 4-D case we have four orthogonal axes x,y,z,w defined by orthonormal vectors $\hat{x}, \hat{y}, \hat{z}, \hat{w}$. These axes make six planes: $xy, xz, xw, yz, yw, zw$. Are these planes orthogonal? For example, the vectors $\hat{z}$ and $\hat{w}$ are perpendicular to the plane $xy$, but they are orthogonal, i.e $\hat{z}\cdot \hat{w}=0$, not parallel. How it is possible that they are not parallel when they are perpendicular to the same plane and how we check if the plane $xy$ is orthogonal to the plane $wz$?