Well the forward vector depends on which coordinate axis you specify as the forward, x,y,z? If y is forward then it should be the second row of a row majour matrix.

Generally engineers have the convention that x and y are on the horisontal plane and z is the vertical, ie z is up and y could very well be forward.

Often in computer graphics however we are heavily influenced by the screen view, where x is left or right y is up and z extends out of the screen.

So in answer, it all depends...

I think the general agreement with vectors is that they represent the axis x, y, and z respectively. So if z is the one you are after then it will be the third row in the matrix.