Given a paralelipiped spanned by $k$ vectors in $\mathbb{R}^n$, $B=\{v_1,\dots, v_k\}$: $$\Omega_B=\{\sum_{i=1}^k \lambda_i v_i|\ 1\geq\lambda_i\geq0 \}$$ I defined the ($k$) volume $\text{Vol}\Omega_B$, recursively by- $$\text{Vol} \Omega_B=\text{Vol}\Omega_{B-\{i\}}\cdot \text{proj}_{\text{Sp}(B-\{i\})^\perp}(v_i)$$
By denoting $A=[v_1\cdots v_k]$ I showed by induction that $\text{Vol}_B=\sqrt{\det(A^TA)}$. Now I define the simplex spanned by these vectors by
$$\Delta_B=\{\sum_{i=1}^k \lambda_i v_i|\ \sum_{i=1}^k \lambda_i=1,\ \lambda_i\geq0 \}$$
Is there a way to derive the ($k$) volume of this shape using what I established prior? It's contained in $\Omega_B$, and after googleing I assume it's actually $\frac{1}{k!}$ of it. I get it for $k=1,2,3$ intuiatively but I can't formally prove it for general $k$.
I don't want to use any integrals, or any relationship between volume under linear transformations and determinants (which as far as I know is derived from the change of variables formula)