Create uncorrelated random multivariate normals Creating correlated random variables is an easy task in R, especially the normal distribution. But what I need is to generate variables (for example 5 standard normals) with very low correlations (e.g., below 0.05). I tried mvtnorm package several times with no success.  
I'm just curious, we use methods (like Cholesky decomposition) to transform uncorrelated variables into correlated. Can we do it the other way around?  
Edit: I want a method that makes multiple "correlated" normal variables "extremely low correlated" without trial & error with a reasonable sample size (say 50) implemented in R. 
 A: Suppose $X$ is multivariate normal. 
Then by definition we can write $X = \mu + CZ$ for a (deterministic) vector $\mu$, matrix $C$ and random vector $Z$ of independent standard normal (i.e. $N(0,1)$) variables.
(So $\Sigma = CC^T$ is the covariance matrix).
Thus if $C$ is invertible then 
$$
C^{-1}(X-\mu) = Z
$$
is a vector of independent (and hence uncorrelated) standard normals.
Note, if you are given pos. def. $\Sigma$ upfront, then to calculate $C = (\Sigma^{1/2})^{-1}$ you would probably want to first calculate the square root (e.g. using Cholesky) and then invert that.
A: Independent random variables should by definition not be correlated, but if you are only generating a small number of points you'll get some correlation by chance. You should find as you generate a larger dataset that you get lower correlations. 
If, for some reason, you need to generate a small dataset with extremely low correlations between your five variables, you might instead generate say 50 variables, calculate their correlation matrix and select 5 with the lowest correlation. 
A: See this, where I borrowed the idea from. I think this is what you were asking for.
# Number of variables to make
n <- 5

# Correlation Matrix with .05 correlation for all
R <- matrix(.05, nrow = n, ncol = n)
diag(R) <- 1

# Cholesky decompostion of correlation matrix
Lut <- chol(R)
L <- t(Lut)

# Standard deviations
sds <- seq(10, 1, length.out = n)

# VCOV matrix
Sigma <- diag(sds) %*% L %*% Lut %*% diag(sds)

# Generate variables
library(MASS)
X <- mvrnorm(50, mu= rep(0, n), Sigma, empirical = TRUE)
cov(X)
cor(X)

X <- mvrnorm(500, mu= rep(0, n), Sigma, empirical = FALSE)
cov(X)
cor(X)

