Notation for cartesian product except one set?

In Wikipedia (https://en.wikipedia.org/wiki/Cartesian_product), I found something, which might be what you are looking for: $\prod_{n=1}^k \Bbb{R} = \Bbb{R}\times \Bbb{R} \times\cdots\times \Bbb{R} = \Bbb{R}^k$. So maybe something like this one is also valid: $$\prod_{\scriptstyle i = 1\atop\scriptstyle i \ne k}^nS_i$$

where $S_i$ is the $i^\text{th}$ set of the list you mentioned.

I have seen a notation for this kind of construction during some of my math lectures (but can't find a reference right now). This was mostly in the context of differential forms (e.g. interior product with vector), but can be applied to your case: $$ S_1\times \dotsm \times \widehat{S_k} \times\dotsm \times S_n := S_1\times \dotsm \times S_{k-1}\times S_{k+1} \times \dotsm S_n$$ The hat denotes the factor to be omitted. Note that this is not a universally standard notation, so even the professors that used it defined it at some point early in the lecture.

In general, we can write

$$S_1 \times \dots \times S_n := \prod_{i=1}^n S_i$$

and then we can apply all conventions we are used to.

As for your question, this can be written as:

$$\prod_{i = 1 \atop i \neq k}^n S_i$$