The determinant in 'n' dimentions is the exterior product of 'n' vectors, this gives the multipier of the pseudoscalar. For instance:
 In 2 dimensions the determinant of 2 vectors is A /\ B which has units of e_{12}
 In 3 dimensions the determinant of 3 vectors is A /\ B /\ C which has units of e_{123}
 In 'n' dimensions the determinant of n vectors is A_{1} /\ ... A_{n} which has units of e_{1...n}
The geometric interpretation of this is shown on this page.
To illustrate this we will start with 2 dimensions:
Two dimensions
Imagine we want to find the directed area enclosed by the parallelogram between two vectors in a plane then we can take the '/\' product:
For example if the two vectors are:
2 e_{1} + 5 e_{2}
3 e_{1} + 4 e_{2}
then the directected area is the bivector part of:
(2 e_{1} + 5 e_{2}) /\ (3 e_{1} + 4 e_{2})
Assuming we are working in euclidean space, that is e_{1} and e_{2} both square to +1, then we get:
2 e_{1} /\ 3 e_{1} + 5 e_{2} /\ 3 e_{1}+ 2 e_{1} /\ 4 e_{2} + 5 e_{2} /\ 4 e_{2}
but the exterior product of common terms is zero: e_{1} /\ e_{1}= e_{2} /\ e_{2} = 0 so the result is:
(2*4  5*3) e_{12}
So the bivector part is 2*4  5*3, in matrix notation this is:

e_{12} 
In the more general case, if we have two vectors (Ax,Ay) and (Bx,By) then the directed area will be (in matrix notation):

= A /\ B 
Note: we can put the e_{1} and e_{2} operators inside this notation:
which expands to give: (Ax e_{1})(By e_{2})  (Ay e_{1})(Bx e_{2}) = (Ax*By  Ay*Bx)e_{12} Order is important here so the determinant is defined as topleft* bottomright  bottomleft *topright 
Can this be generalised to 'n' dimensions?

= Ax 

 Ay 

+ Az 

and recursing down to the next level gives:
M = Ax By Cz + Ay Bz Cx + Az Bx Cy  Ax Bz Cy  Ay Bx Cz  Az By Cx
A /\ B /\ C = (Ax e_{1} + Ay e_{2}+ Az e_{3}) /\ (Bx e_{1} + By e_{2}+ Bz e_{3}) /\ (Cx e_{1} + Cy e_{2}+ Cz e_{3})
= (Ax*By e_{12}+ Ax*Bz e_{13}
 Ay*Bx e_{12} + Ay*Bz e_{23}
+ Az*Bx e_{31}  Az*By e_{23}) /\ (Cx e_{1} + Cy e_{2}+ Cz e_{3})
= ((Ax*By Ay*Bx) e_{12}+ (Az*BxAx*Bz) e_{31}
+ (Ay*Bz  Az*By) e_{23})) /\ (Cx e_{1} + Cy e_{2}+ Cz e_{3})
= (Ax*By Ay*Bx)e_{12} (Cx e_{1} + Cy e_{2}+ Cz e_{3})
+ (Az*BxAx*Bz)e_{31} (Cx e_{1} + Cy e_{2}+ Cz e_{3})
+ (Ay*Bz  Az*By) e_{23})(Cx e_{1} + Cy e_{2}+ Cz e_{3})
= (Ax*By Ay*Bx)Cz e_{123} + (Az*BxAx*Bz)Cy e_{312} + (Ay*Bz  Az*By)Cx e_{231}
so the resulting pseudoscalar term is:
Ax*By*Cz Ay*Bx*Cz + Az*Bx*CyAx*Bz*Cy+ Ay*Bz*Cx  Az*By*Cx
which has the same form as M above with a few sign changes, I must have reversed the order somewhere.