The declaration of a field as a triplet $(\mathbb{K},\oplus,\odot)$ is canonical and in every book I've read, the definition of field is written using it. I can use this notation to describe any field in a strict and technical way. For example, this is how I can describe the finite field $\Bbb{Z}_2$:
$(\Bbb{Z}_2:=\{0,1\},\\\ \oplus:\Bbb{Z}_2^2\to\Bbb{Z}_2:=(x+y)\mod2,\\\ \odot:\Bbb{Z}_2^2\to\Bbb{Z}_2:=(x\cdot y)\mod2)$
I'm looking for a likewise technically strict way to describe a vector space over a field, since I've only found verbose specifications in common language (e.g., "a vector space $\mathbb{V}$ with operations $\oplus$ and $\odot$ over a field $\mathbb{k}$ with operations $+$ and $\cdot$").
There is any formal notation in which I can specify a vector space in few words (actually, in no words at all) as I do with fields?