Is there way to write the following integral representations of integer zeta values in a general form?

Question

How do you write this as a general formula for all integer zeta values greater than $0$?

$$\zeta(1)=\int_0^1 \frac{1}{1-x}dx$$ $$\zeta(2)=\int_0^1\int_0^1\frac{1}{1-xy}dxdy$$ $$\zeta(3)=\int_0^1\int_0^1\int_0^1\frac{1}{1-xyz}dxdydz$$

$$\zeta(n) = ?$$

Answer 1

$$\zeta(n) = \int_{[0,1]^n} \dfrac{1}{1-\prod_{i=1}^n x_i} dx_1 \ldots dx_n$$