Will normal random variables form a normal random vector? If $X_1, ..., X_n$ are normal random variables, will $[X_1, ..., X_n]$ be a normal random vector?
I know if $X_1, ..., X_n$  are further independent, then $[X_1, ..., X_n]$ is a normal random vector. Can there be more relaxed condition on $X_1, ..., X_n$ than being independent?
Thanks in advance!

$[X_1,...,X_n]$ is a normal (or Gaussian) random vector is the same as saying that those variables are jointly normal -or that it is a multivariate Gaussian. That each $X_i$ is normal (i.e., the marginal distributions are normal) is necessary but not sufficient (easy to give counterexamples). That each $X_i$ is normal and they are independent is -as you say- sufficient, but not necessary. A general (necessary and sufficient) condition can be expressed in terms of a linear combination of normal iid scalar variables.
Specifically: given $Z_1 ... Z_n$ iid standard normals (mean 0 and variance 1), then  $X = A Z + b$  (with A any square nonsingular matrix and b any fixed vector) is jointly normal - and this is fully general. From this comes the formula of the (general) multivariate Gaussian variable. Conversely, if $X$ is a normal random vector then one can find a matrix C and a vector d such that $Z = C X + d$ is a vector of iid standard normal variables. This comes to be a multivariate generalization of the well known formula to standardize a Gaussian: $z = \frac{x-\mu}{\sigma}$
