Normalization conventions for explicit tensor representations of a $k$-form

Question

There are all kinds of confusing combinatorial factors that crop up in the exterior algebra, especially if you're trying to work with explicit array representations rather than abstract objects. I've consulted several references, but can't find the normalizations listed in one place. I think I have all the normalizations correct, but could someone please verify? (Claim #6 looks weird to me.)

In all cases $A_{i_1 \dots i_k}$ is a totally antisymmetric explicit tensor component representation of a $k$-vector element of $\Lambda^k(\mathbb{R}^n)$, the $k$th exterior power of $\mathbb{R}^n$. $\{\hat{e}_i\}$, $i = 1,\dots, n$ represents an orthonormal basis for $\mathbb{R}^n$. I'm using the standard inner product on $\Lambda^k(\mathbb{R}^n)$ that linearly extends the formula $$ \langle v_1 \wedge \dots \wedge v_k, u_1 \wedge \dots \wedge u_k \rangle_{\Lambda^k(\mathbb{R}^n)} := \det\left( \langle u_i, v_j \rangle \right). $$

(I'm using the "geometer's" normalization convention that if $A$ is a $k$-form and B is an $l$-form, then $A \wedge B = \frac{(p+q)!}{p! q!} \mathrm{Alt}(A \otimes B)$, not the "algebraist's" convention $A \wedge B = \mathrm{Alt}(A \otimes B)$.)

1. An exterior algebra basis vector $\hat{e}_{i_1} \wedge \dots \wedge \hat{e}_{i_k}$ has norm 1. Its tensor representation has entries $\{1, 0, -1\}$ with no overall normalization constant.

2. In terms of this standard tensor representation, a generic $k$-vector $A$ has norm-squared $$|A|^2 = \frac{1}{k!} A_{i_1 \dots i_k} A^{i_1 \dots i_k}$$ with no dependence on $n$ in the normalization constant.

3. The top-vector $$\hat{e}_1 \wedge \dots \wedge \hat{e}_n = \epsilon_{i_1 \dots i_n} dx^{i_1} \otimes \dots \otimes dx^{i_n} = \frac{1}{n!} \epsilon_{i_1 \dots i_n} dx^{i_1} \wedge \dots \wedge dx^{i_n}$$ has norm 1.

4. A contraction of $A$ and $B$ over $p$ different indices yields the tensor $$\frac{1}{p!} A_{i_1 \dots i_p j_{p+1} \dots j_k} B^{i_1 \dots i_p l_{p+1} \dots l_m}$$

5. The Hodge dual $\star A$ of a $k$-vector $A$ has array representation $$ (\star A)_{i_1 \dots i_{n-k}} = \frac{1}{k!} \epsilon_{i_1 \dots i_{n-k}}^{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ j_1 \dots j_k} A_{j_1 \dots j_k}, $$ with no dependence on $n$ in the normalization constant.

6. $$\star \left( \hat{e}_{i_1} \wedge \dots \wedge \hat{e}_{i_k} \right) = \frac{1}{(n-k)!} \epsilon_{i_1 \dots i_k}^{\ \ \ \ \ \ \ \ \ \ \ j_1 \dots j_{n-k}} \hat{e}_{j_1} \wedge \dots \wedge \hat{e}_{j_{n-k}},$$ with a normalization constant that depends on $n$. (I'm not sure if this is consistent with #5 above, but the fact that we're wedging together instead of tensoring together basis vectors may change the combinatorial factors.)

7. The Hodge star operator preserves the norm. That is, $$|A|_{\Lambda^k(\mathbb{R}^n)} \equiv |\star A|_{\Lambda^{n-k}(\mathbb{R}^n)}.$$

Are these normalization factors are mutually consistent, and if so, are they standard under the "geometer's" convention?