Recording 2 examples here.

The first is to consider for n > 1 the map

A^n —> P^n, (x_1, …, x_n) maps to (x_1x_2…x_n : x_1 – 1 : … : x_n – 1)

This map is quasi-finite and flat, but it is not surjective as the points (1:1:0…0), (1:0:1:0…0), …, (1:0…0:1) are missing in the image. If we take as homogeneous coordinates on P^n the variables T_0, …, T_n then the inverse image of T_1 + … + T_n = 0 is the hyperplane x_1 + … + x_n = n in A^n. Thus we see

There is a surjective quasi-finite flat morphism A^{n – 1} —> P^{n – 1}.

The map we constructed has degree n and that is also the minimum possible.

The second example is to consider for n > 1 the map

A^n —> A^n – {0}, (x_1,…,x_n) maps to (x_1, …, x_{n – 2}, x_{n – 1}x_n – 1, f)

where

f = x_1x_{n – 1}^{n – 1} + … + x_{n – 3}x_{n – 1}^3 + x_{n – 2}x_{n – 1}^2 + x_{n – 1}(x_{n – 1}x_n – 1) + x_n

This map is surjective, quasi-finite flat of degree n (and again that’s minimal).

Enjoy!