Making a box from a transform

Hi. I'm trying to make a Box's normals in code that is realigned based on a given "up vector." The thing is, I don't really know the math that would be needed to do this. Since, it's hard for me to explain this, example: a box is setup in code like:

Point3F BoxNormals[] = {
Point3F(-1, 0, 0),
Point3F( 1, 0, 0),
Point3F( 0,-1, 0),
Point3F( 0, 1, 0),
Point3F( 0, 0,-1),
Point3F( 0, 0, 1)
};

I'd like to be able to use that to make normals for the sides of a box, but somehow transform them so the "up" for the box, is equal to the up vector, and the sides would have to turn to match. Example:

Point3F upVector = //however the upVector comes into this equation
Point3F BoxNormals[] = {
Point3F(x,y,z),
Point3F(x,y,z),
Point3F(x,y,z),
Point3F(x,y,z),
upVector, //These 2 are easy, since it's the top/bottom of the box
-upVector,
//etc...
};

I'm good at 2d/3d math, and know all the terms so don't feel you need to explain things much. I just can't figure out how I would do this, and searching turns up no related results. Thanks!