What result do you expect, if you are not providing the actual matrix input?

Make your matrix

mm[n_] :=  Table[If[i == 1, x^j, a[i - 1, i]^j], {i, 1, n}, {j, 0, n - 1}]
mm[3] // MatrixForm

and calculate its determinant

det[m_] := Product[
  (m[[j, 2]] - m[[i, 2]]), {i, 1, Length[m] - 1}, {j, i + 1,  Length[m]}]

Example

det[mm[5]] - Det[mm[5]] // Expand

The det-method is (of course) considerably faster

Timing[det[mm[10]];]
Timing[Det[mm[10]];]

It's a Vandermonde matrix. The determinant is the product of differences between elements in the second column (subtracting lower-row elements from higher.)

I am afraid there is no simple closed expression for the determinant:

mm[n_] :=  Table[If[i == 1, x^j, a[i - 1, j]^j], {i, 1, n}, {j, 0, n - 1}]
mm[3] // MatrixForm
Collect[Det[mm[4]], x]