Pick two linearly-independent vectors w,v which each have zero dot product with [1,1,1]. Then (w,v) is a basis for the plane perpendicular to [1,1,1], and (w,v,[1,1,1]) is a basis for R3. Then projection onto the line L sends w and v to zero, and [1,1,1] to itself, which is all you need to know to write its matrix under that basis.